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Radical Equation Solver

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Radical Calculator

Instructions: Use radical calculator to compute and simplify any expression involving radicals that you provide, showing all the steps. Please type in the radical expression you want to work out in the form box below.

About this Radical Calculator

This calculator is a radical calculator with steps that will allow you to calculate and simplify a given radical expression you provide, showing all the steps of the process, or if the expression is already simplified, the calculator will tell you so.

So then, once the required expression has been typed in, all you need to do is click on the "Calculate" button to get all the relevant steps shown to you.

Radical expressions are very common in Algebra, especially when using square root. Other radical expressions are usually expressed in terms of the calculation of exponents , because of ease of notation.

How to simplify radical expressions?

Radical expressions are more than just dealing with square roots. There is usually some familiarity with operations involving square roots, like:

for example. Most people would not have a problem with coming up with the right answer, which would be

With radicals though, it is a little harder to interpret, for example, when you need to simplify something like \(\sqrt[3]{x^4}\). Usually the best strategy would be to express radicals in their equivalent power expression , where the exponents would do the job.

Specifically, we know that by definition: \(\sqrt[n]{a} = a^{1/n}\), which gives a DIRECT link between radicals and powers.

Steps for dealing with a radical form

• Step 1: Clearly identify the expression you want to calculate or simplify
• Step 2: Identify the type of root you have. Is it a basic square root, or is it another radical?
• Step 3: If you have a square root, chances are that simplification could be relatively easy, for which you can use this calculator
• Step 4: For other types of radicals, you want to use the fact that \(\sqrt[n]{a} = a^{1/n}\) holds, and perhaps an appropriate strategy consists of expressing all the radicals as powers, and use the rule of exponents to simplify the expression

In practice, dealing with radical forms can be cumbersome, and even confusing, reason which converting all radicals to exponents and use the rules of exponents may be the best way to go.

Using this Radical Calculator

Using this calculator is simple, all you need to do is to correctly specify an expression that involves a radical, it could be a square root like 'sqrt(4x^2)', or it could be a more complex radical like 'sqrt[5](128x^8)'

You need to just type an expression that system recognizes as valid. Often times you can type directly the radical form in terms of exponents, like '(81 x^4)^(1/3)'.

Why you should use a radical simplifier?

Because radical expressions appear very frequently in all realms of Algebra and Calculus. Usually, when solving equations you run into the need to dealing with radical forms to find a simpler form of an equation solution.

There are often times where you will need help with dividing radicals and showing steps, for which case the use of a calculator like this could prove invaluable. Mostly it is about time saving and double checking your own answers.

Example: Calculating the sum of fractions

Simplify the following radical form: \(\sqrt[3]{81x^4 y^7}\)

Solution: In this case, it would be useful to express the radical term provided using exponents instead of radicals:

Now, we rewrite the internal terms and simplify:

which concludes the calculation.

Other useful Algebra calculators

This calculator will be useful to you whenever you need to simplify a radical form. But there are lots of other types of expressions that you could potentially encounter.

If you need help with simplifying a fraction or doing a general simplification of an expression , calculators that show steps can make the whole difference..

There are different and multiple connections between different types of math objects, links between decimals and radicals, decimals and fractions , a too many to count.

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Radical Equation Calculator + Online Solver With Free Steps

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What Is the Radical Equation Calculator?

Higher-order polynomials.

\[ \text{radical equation} : \sqrt[n]{\text{variable terms}} + \text{other terms} = 0 \]

\[ \sqrt{5x^2+10x}+4x-7 = 0 \]

The calculator supports multi-variable equations , but the intended usage is for single-variable ones . That is because the calculator accepts only one equation at a time and cannot solve systems of simultaneous equations where we have n equations with m unknowns. 

Thus, for multi-variable equations, the calculator outputs roots in terms of the other variables.

The Radical Equation Calculator is an online tool that evaluates the roots for a given radical equation representing a polynomial of any degree and plots the results. 

The calculator interface consists of a single text box labeled “Equation.” It is self-explanatory – you enter the radical equation to solve here. You may use any number of variables, but, as mentioned earlier, the intended usage is for single-variable polynomials of any degree.

How To Use the Radical Equation Calculator?

You can use the Radical Equation Calculator by entering the given radical equation into the input text box. For example, suppose you want to solve the equation:

\[ 7x^5 +\sqrt{6x^3 + 3x^2}-2x-4 = 0 \]

Then you can use the calculator by following the step-by-step guidelines below.

Enter the equation in the text box. Enclose the radical term in “sqrt(radical term)” without quotes. In the example above, you would enter “7x^5+sqrt(6x^3+3x^2)-2x-4=0” without quotes. 

Note: Do not enter just the side of the equation with the polynomial! Otherwise, the results will not contain the roots.

Press the Submit button to get the results.

The result section primarily consists of:

• Input: The calculator’s interpretation of the input equation. Useful for verifying the equation and ensuring that the calculator handles it correctly.
• Root Plots: 2D/3D plots with the roots highlighted. If at least one of the roots is complex, the calculator additionally draws them on the complex plane. 
• Roots/Solution: These are the exact values of the roots. If they are a mixture of complex and real values, the calculator shows them in the separate sections “Real Solutions” and “Complex Solutions.”

There are also a couple of secondary sections (possibly more for different inputs):

• Number Line: The real roots as they fall onto the number line.
• Alternate Forms: Various rearrangements of the input equation.

For the example equation , the calculator finds a mixture of real and complex roots:

\[ x_{r} \approx 0.858578  \]

\[ x_{c_1,\,c_2} \approx 0.12875 \pm 0.94078i \qquad x_{c_3,\,c_4} \approx -0.62771 \pm 0.41092i \]

How Does the Radical Equation Calculator Work?

The Radical Equation Calculator works by isolating the radical term on one side of the equation and squaring both sides to remove the radical sign. After that, it brings all the variable and constant terms to one side of the equation, keeping 0 on the other end. Finally, it solves for the roots of the equation, which is now a standard polynomial of some degree d.

The calculator can quickly solve for polynomials with degrees greater than four. That is significant because there is no general formulation for solving d-degree polynomials with d > 4. 

Extracting the roots of these higher-order polynomials requires a more advanced method such as the iterative Newton method. By hand, this method takes a long time because it is iterative, requires initial guesses, and may fail to converge for certain functions/guesses. However, this is not a problem for the calculator!

Solved Examples

We will stick to lower-order polynomials in the following examples to explain the basic concept since solving higher-order polynomials with the Newton method will take a lot of time and space.

Consider the following equation:

\[ 11 + \sqrt{x-5} = 5 \] 

Calculate the roots if possible. If not possible, explain why.

Isolating the radical term:

\[ \begin{aligned} \sqrt{x-5} &= 5-11 \\ &= -6 \end{aligned} \]

Since the square root of a number cannot be negative, we can see that no solution exists for this equation. The calculator verifies this as well.

Solve the following equation for y in terms of x.

\[ \sqrt{5x+3y}-3 = 0 \]

Isolating the radicals:

\[ \sqrt{5x+3y} = 3 \]

Since this is a positive number, we are safe to proceed. Squaring both sides of the equation:

\[ 5x+3y = 3^2 = 9 \]

Rearranging all terms to one side:

5x+3y-9 = 0 

It is the equation of a line! Solving for y:

Dividing both sides by 3:

\[ y = -\frac{5}{3}x + 3 \]

The y-intercept of this line is at 3. Let us verify this on a graph:

The calculator also provides these results. Note that as we had only one equation, the solution is not a single point. It is constrained to a line instead. Similarly, if we had three variables instead, the set of possible solutions would lie on a plane!

Find the roots for the following equation:

\[ \sqrt{10x^2+20x}-3 = 0 \]

Separating the radical term and squaring both sides after:

\[ \sqrt{10x^2 + 20x} = 3 \]

\[ 10x^2 + 20x = 9 \, \Rightarrow \, 10x^2+20x-9 = 0 \]

That is a quadratic equation in x. Using the quadratic formula with a = 10, b = 20, and c = -9:

\begin{align*} x_1,\, x_2 & = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\\\ & = \frac{-20 \pm \sqrt{20^2-4(10)(-9)}}{2(10)} \\\\ & = \frac{-20 \pm \sqrt{400+360}}{20} \\\\ & = \frac{-20 \pm \sqrt{760}}{20} \\\\ & = \frac{-20 \pm 27.5681}{20} \\\\ & = -1 \pm 1.3784 \end{align*}

We get the roots:

\[ \therefore , x_1 = 0.3784 \quad , \quad x_2 = -2.3784 \]

The calculator outputs the roots in their exact form:

\[ x_1 = -1 + \sqrt{\frac{19}{10}} \approx 0.3784 \quad,\quad x_2 = -1-\sqrt{\frac{19}{10}} \approx -2.3784 \]

The plot is below:

Consider the following radical with nested square roots:

\[ \sqrt{\sqrt{x^2-4x}-9x}-6 = 0 \]

Evaluate its roots.

First, we isolate the outer radical as usual:

\[ \sqrt{\sqrt{x^2-4x}-9x} = 6 \]

Squaring both sides:

\[ \sqrt{x^2-4x}-9x = 36 \]

Now we need to remove the second radical sign as well, so we isolate the radical term again:

\[ \sqrt{x^2-4x} = 9x+36 \]

\[ x^2-4x = 81x^2+648x+1296 \]

\[ 80x^2+652x+1296 = 0 \]

Dividing both sides by 4:

\[ 20x^2+163x+324 = 0 \]

Solving using the quadratic formula with a = 20, b = 163, c = 324:

\begin{align*} x_1,\, x_2 & = \frac{-163 \pm \sqrt{163^2-4(20)(324)}}{2(20)} \\\\ & = \frac{-163 \pm \sqrt{26569 – 25920}}{40} \\\\ &= \frac{-163 \pm \sqrt{649}}{40} \\\\ & = \frac{-163 \pm 25.4755}{40} \\\\ & = -4.075 \pm 0.63689 \end{align*}

\[ \therefore \,\,\, x_1 = -3.4381 \quad , \quad x_2 = -4.7119 \]

However, if we plug in $x_2$ = -4.7119 into our original equation, the two sides are not equal:

\[ 6.9867-6 \neq 0 \]

Whereas with $x_1$ = -3.4381, we get:

\[ 6.04-6 \approx 0 \]

The slight error is due to the decimal approximation. We can verify this in the figure as well:

All graphs/images were created with GeoGebra.

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Radical Calculator

Do you want to get the solution of radical expression? Not a problem as the radical calculator is here to give you accurate solution of radicals problems for free.

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Introduction to Radical Calculator

Radical Calculator is an online tool that helps you find the solution to radical expression in simpler form in a few seconds. Our tool helps to determine whether the radical number is completely reduced from an algebraic expression or not.

When you manually calculate radical numbers, you might need clarification because some expressions do not easily give solutions because of their complex behavior. To get rid of all the problems you need our simplifying radicals calculator that gives you solutions to any type of radical problem.

What is a Radical

Radical is any number that has a root number in a mathematical expression. The root may vary from square, cube, fourth, or nth root. Radical is known as the inverse of the exponential method. It is used when you want to convert a radical expression into a simplified term in solution.

The power of radical that is written outside is called the index which may be from 1 to nth root. It can be represented with a square root or radical symbol as “√ “.The number written inside the radical is called radicand.

Formula for Radical Expression

A radical expression or formula of radical expression can be represented as ⁿ√x. Where " a " is the constant , that is used for the root n=1,2,3,4…. and"x" is the radicand or the number under the radical sign,

$$ \sqrt[n]{X} $$

How to Calculate Radical Expressions

The radical method can be used for reducing the index or root number from the given expression into the simplest form. You can use different mathematical operations like multiplying and dividing with conjugate or finding factors to eliminate the root (square, cube, fourth,...nth).

For the simplification of a radical expression, we can eliminate the n th root into its simplified form. To perform calculations involving radical expressions, you can use a radical calculator.

For example the radical in the form a ⁿ√x. The steps given below that used to simplify a radical expression:

Consider an expression n under the radical sign as ⁿ√x.

Find the product of x as a prime factor.

To reduce the nth root, write all prime factors in a group that have the same value in powers of n.

Now write all the factors into its exponential power as n th power, it can be written outside the radical sign.

It multiplies all remaining terms under the radical sign to simplify the radical expression. If all terms under the radical sign are reduced, it means it changes the radical number into the simplest form.

Solved Example of Radical Expression

An example of a radical expression problem along with its solution is given to understand the manual calculations. As our radical form calculator can solve such problems easily but it is important to know the manual way as well.

Find the radical expression √50?

$$ Factor\; 50 $$

$$ 50 \;=\; 2 \times 25 \;=\; 2 \times 5^2 $$

Identify and pull out the perfect square factors,

$$ \sqrt{50} \;=\; \sqrt{2 \times 5^2} $$

Pull out 5 from:

$$ 5^2: 5 \times \sqrt{2} $$

Multiply the simplified expression,

$$ 5 \times \sqrt{2} \;=\; \sqrt[5]{2} $$

How to Use the Radical Calculator

Radical Solver has a simple design tool that enables you to use it to calculate algebraic expressions with different mathematical operations.

Before entering the input into the multiplying radicals calculator, you must follow some simple steps so that you get a smooth experience during the calculation. These steps are:

• Enter the number used in the radical expression in the input box.
• Enter the index or root number in the input box.
• Review your given number before hitting the calculate button to start the evaluation process.
• Click the “Calculate” button to get the result of your given radical expression problem.
• If you want to try out our radical equation calculator for the first time then you can use the load example to get a better understanding of the concept of algebraic expression simplification.
• Click on the “Recalculate” button to get a refresh page for more solutions to radical problems

Final Result of Simplifying Radicals Calculator

Radical calculator gives you the solution to a given radical question when you add the input into it. It provides you with solutions with a detailed procedure instantly. It may be included as:

• Result Option

When you click on the Result option it gives you a solution to find the given nth root problem

• Possible Steps

It provides you with a solution in which all the evaluation processes are present in a step-by-step method when you click on this option.

Advantages of the Radical Solver

Simplifying radicals calculator provides you with multiple benefits whenever you use it to calculate radical problems and gives you a solution. These benefits are:

• Radical form calculator is a free-of-cost tool that enables you to use it anytime to find the algebraic expression without any fee.
• Multiplying radicals calculator is an adaptable tool that allows you to get the solution of various kinds of radical expression problems
• You can try out our tool to practice more examples of the radical questions so that you get more familiar with this concept
• Our radical calculator saves the time that you spend on doing the Radical expression calculations manually because it gives you solutions in a run of time.
• Radical Equation Calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate root number examples without any mistakes.
• It provides the solution with a complete process in a stepwise method so that you get clarity on the Radical problems

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Tiger Algebra Calculator

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• Factoring binomials as difference of squares
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Radical Calculator – Simplify Radicals & Square Roots

This tool will calculate the square root of any number you input.

How to Use the Radical Calculator

To use this radical calculator, enter the number you wish to find the root of (the radicand) and the degree of the root in the provided input fields. The degree of the root must be a non-zero integer. After entering the values, click the “Calculate” button to see the result.

How Results are Calculated

The calculator takes the given radicand and the degree of the root and computes the result using the mathematical formula for roots. If the degree of the root is positive, it calculates it as the power of the radicand raised to the (1/degree). If the degree is negative, it calculates it as the radicand raised to the (-1/absolute degree).

Limitations

This calculator has the following limitations:

• The degree of the root must be a non-zero integer.
• Inputs must be numeric and valid numbers.
• Accuracy might be limited by the floating-point precision of JavaScript.

Use Cases for This Calculator

Finding square roots.

When you need to calculate the square root of a number, a radical calculator is your perfect companion. Simply input any positive number, and the tool will swiftly deliver the accurate square root, simplifying your mathematical tasks.

Solving Quadratic Equations

Quadratic equations can be tricky, but with a radical calculator, you can easily find their solutions. By using the quadratic formula, you can input the coefficients and get the real roots presented clearly and quickly.

Evaluating Radical Expressions

Are you struggling with complex radical expressions? The radical calculator allows you to evaluate and simplify these expressions seamlessly, ensuring you understand the results without getting overwhelmed by the math.

Converting Radicals to Decimal Form

Sometimes, you need your answers in decimal form rather than in radical notation. The radical calculator can effortlessly convert any radical expression into its decimal equivalent, making your results more accessible and easier to interpret.

Performance of Addition and Subtraction Operations

Addition and subtraction of radical expressions can be confusing without the right tools. With a radical calculator, you can combine these expressions with ease, providing you with the correct simplified outcome in no time.

Understanding Exponents with Radicals

Do you want to dive deeper into the relationship between exponents and radicals? The radical calculator helps you manipulate and analyze expressions involving both, enhancing your comprehension of these fundamental mathematical concepts.

Graphing Radical Functions

Visualizing the graph of a radical function can be incredibly beneficial for understanding its properties. You can use a radical calculator to evaluate specific points on the graph, giving you an accurate representation of how the function behaves.

Simplifying Complex Numbers

Complex numbers often include radicals, which can be challenging to simplify. A radical calculator streamlines the process, allowing you to simplify expressions that involve imaginary numbers effectively and confidently.

Applications in Physics and Engineering

Radicals often appear in various physics and engineering calculations. Whether you’re dealing with formulas related to motion or energy, the radical calculator can simplify your computations, helping you focus on problem-solving rather than calculations.

Learning Tool for Students

If you’re a student trying to grasp the concepts of radicals, a radical calculator serves as an excellent learning aid. It not only provides answers but also shows step-by-step solutions, helping you understand the underlying processes involved in calculations.

Other Resources and Tools

• Algebra 2 Calculator – Accurate Solutions
• Equation Calculator – Solve Math Problems Easily
• Algebrator Calculator – Solve Algebra Problems Easily
• Quadratic Function Calculator – Solve Equations Effortlessly
• Root Calculator – Fast & Accurate Root Calculations

Radical Equations

Solving radical equations.

Learning how to solve radical equations requires a lot of practice and familiarity of the different types of problems. In this lesson, the goal is to show you detailed worked solutions of some problems with varying levels of difficulty.

What is a Radical Equation?

An equation wherein the variable is contained inside a radical symbol or has a rational exponent. In particular, we will deal with the square root which is the consequence of having an exponent of [latex]\Large{1 \over 2}[/latex].

Key Steps to Solve Radical Equations:

1) Isolate the radical symbol on one side of the equation

2) Square both sides of the equation to eliminate the radical symbol

3) Solve the equation that comes out after the squaring process

4) Check your answers with the original equation to avoid extraneous values or solutions

Examples of How to Solve Radical Equations

Example 1 : Solve the radical equation

The radical is by itself on one side so it is fine to square both sides of the equations to get rid of the radical symbol. Then proceed with the usual steps in solving linear equations.

You must ALWAYS check your answers to verify if they are “truly” the solutions. Some answers from your calculations may be extraneous. Substitute x = 16 back into the original radical equation to see whether it yields a true statement.

Yes, it checks, so x = 16 is a solution.

Example 2 : Solve the radical equation

The setup looks good because the radical is again isolated on one side. So I can square both sides to eliminate that square root symbol. Be careful dealing with the right side when you square the binomial (x−1). You must apply the FOIL method correctly.

We move all the terms to the right side of the equation and then proceed with factoring out the trinomial. Applying the Zero-Product Property, we obtain the values of x = 1 and x = 3 .

Caution: Always check your calculated values from the original radical equation to make sure that they are true answers and not extraneous or “false” answers.

Looks good for both of our solved values of x after checking, so our solutions are x = 1 and x = 3 .

Example 3 : Solve the radical equation

We need to recognize the radical symbol is not isolated just yet on the left side. It means we have to get rid of that −1 before squaring both sides of the equation. A simple step of adding both sides by 1 should take care of that problem. After doing so, the “new” equation is similar to the ones we have gone over so far.

Our possible solutions are x = −2 and x = 5 . Notice I use the word “possible” because it is not final until we perform our verification process of checking our values against the original radical equation.

Since we arrive at a false statement when x = -2, therefore that value of x is considered to be extraneous  so we disregard it! Leaving us with one true answer, x = 5 .

Example 4 : Solve the radical equation

The left side looks a little messy because there are two radical symbols. But it is not that bad! Always remember the key steps suggested above. Since both of the square roots are on one side that means it’s definitely ready for the entire radical equation to be squared.

So for our first step, let’s square both sides and see what happens.

It is perfectly normal for this type of problem to see another radical symbol after the first application of squaring. The good news coming out from this is that there’s only one left. From this point, try to isolate again the single radical on the left side, which should force us to relocate the rest to the opposite side.

As you can see, that simplified radical equation is definitely familiar . Proceed with the usual way of solving it and make sure that you always verify the solved values of x against the original radical equation.

I will leave it to you to check that indeed x = 4 is a solution.

Example 5 : Solve the radical equation

This problem is very similar to example 4. The only difference is that this time around both of the radicals has binomial expressions. The approach is also to square both sides since the radicals are on one side, and simplify. But we need to perform the second application of squaring to fully get rid of the square root symbol.

The solution is x = 2 . You may verify it by substituting the value back into the original radical equation and see that it yields a true statement.

Example 6 : Solve the radical equation

It looks like our first step is to square both sides and observe what comes out afterward. Don’t forget to combine like terms every time you square the sides. If it happens that another radical symbol is generated after the first application of squaring process, then it makes sense to do it one more time. Remember, our goal is to get rid of the radical symbols to free up the variable we are trying to solve or isolate.

Well, it looks like we will need to square both sides again because of the newly generated radical symbol. But we must isolate the radical first on one side of the equation before doing so. I will keep the square root on the left, and that forces me to move everything to the right.

Looking good so far! Now it’s time to square both sides again to finally eliminate the radical.

Be careful though in squaring the left side of the equation. You must also square that −2 to the left of the radical.

What we have now is a quadratic equation in the standard form. The best way to solve for x is to use the Quadratic Formula where a = 7, b = 8, and c = −44.

So the possible solutions are [latex]x = 2[/latex], and [latex]x = {{ – 22} \over 7}[/latex].

I will leave it to you to check those two values of “x” back into the original radical equation. I hope you agree that x = 2 is the only solution while the other value is an extraneous solution, so disregard it!

Example 7 : Solve the radical equation

There are two ways to approach this problem. I could immediately square both sides to get rid of the radicals or multiply the two radicals first then square. Both procedures should arrive at the same answers when properly done. For this, I will use the second approach.

Next, move everything to the left side and solve the resulting Quadratic equation.  You can use the Quadratic formula to solve it, but since it is easily factorable I will just factor it out.

The possible solutions then are [latex]x = {{ – 5} \over 2}[/latex] and [latex]x = 3[/latex] .

I will leave it to you to check the answers. The only answer should be [latex]x = 3[/latex] which makes the other one an extraneous solution.

You might also like these tutorials:

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Solving Radical Equations

How to solve equations with square roots, cube roots, etc.

Radical Equations

We can get rid of a square root by squaring (or cube roots by cubing, etc).

Warning: this can sometimes create "solutions" which don't actually work when we put them into the original equation. So we need to Check!

Follow these steps:

• isolate the square root on one side of the equation
• square both sides of the equation

Then continue with our solution!

Example: solve √(2x+9) − 5 = 0

Now it should be easier to solve!

Check: √(2·8+9) − 5 = √(25) − 5 = 5 − 5 = 0

That one worked perfectly.

More Than One Square Root

What if there are two or more square roots? Easy! Just repeat the process for each one.

It will take longer (lots more steps) ... but nothing too hard.

Example: solve √(2x−5) − √(x−1) = 1

We have removed one square root.

Now do the "square root" thing again:

We have now successfully removed both square roots.

Let us continue on with the solution.

It is a Quadratic Equation! So let us put it in standard form.

Using the Quadratic Formula (a=1, b=−14, c=29) gives the solutions:

2.53 and 11.47 (to 2 decimal places)

Let us check the solutions:

There is really only one solution :

Answer: 11.47 (to 2 decimal places)

See? This method can sometimes produce solutions that don't really work!

The root that seemed to work, but wasn't right when we checked it, is called an "Extraneous Root"

So checking is important.

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Solving Equations with Radicals

A "radical" equation is an equation in which there is a variable inside the radical sign

Four steps to solve equations with radicals

Step 1: Isolate the radicals to left side of the equal sign.

Step 2: Square each side of the equation

Step 3: Solve the resulting equation

Step 4: Check all solutions

Equations with one radical

Example 1: Solve $\sqrt {2x + 3} - x = 0$

$$\sqrt {2x + 3} - x = 0$$ $$\sqrt {2x + 3} = x$$

$${\left( {\sqrt {2x + 3} } \right)^2} = {(x)^2}$$ $$2x + 2 = {x^2}$$ $${x^2} - 2x - 3 = 0$$

$${x^2} - 2x - 3 = 0$$ $$a = 1, b = - 2, c = - 3$$ $${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$$ $${x_{1,2}} = \frac{{ - ( - 2) \pm \sqrt {{{( - 2)}^2} - 4 \cdot 1 \cdot ( - 3)} }}{{2 \cdot 1}}$$ $${x_{1,2}} = \frac{{2 \pm \sqrt {4 + 12} }}{2}$$ $${x_{1,2}} = \frac{{2 \pm 4}}{2}$$ $${x_1} = \frac{{2 + 4}}{2},{x_2} = \frac{{2 - 4}}{2}$$ $${x_1} = 3,{x_2} = - 1$$

Let's check to see if x 1 = 3 is solution:

$$\sqrt {2x + 3} - x = 0$$ $$\sqrt {2 \cdot 3 + 3} - 3 = 0$$ $$\sqrt 9 - 3 = 0$$ $$3 - 3 = 0$$ $$0 = 0, OK$$

Let's check to see if x 2 = -1 is solution:

$$\sqrt {2x + 3} - x = 0$$ $$\sqrt {2 \cdot ( - 1) + 3} - ( - 1) = 0$$ $$\sqrt { - 2 + 3} + 1 = 0$$ $$\sqrt 1 + 1 = 0$$ $$2 = 0,NOTOK$$

So, the original equation had a single solution x = 3 .

Exercise 1: Solve equations

Example 2: Solve $\sqrt {4x + 3} + 2x - 1 = 0$

$$\sqrt {4x + 3} + 2x - 1 = 0$$ $${(\sqrt {4x - 3} )^2} = {(1 - 2x)^2}$$ $$4x - 3 = 1 - 4x + 4{x^2}$$ $$4x - 3 - 1 + 4x - 4{x^2} = 0$$ $$ - 4{x^2} + 8x - 4 = 0/:( - 4)$$ $${x^2} - 2x + 1 = 0$$ $${(x - 1)^2} = 0$$ $$x = 1$$

Let's check it to see if x = 1 is a solution to the original equation.

$$\sqrt {4x - 3} + 2x - 1 = 0$$ $$\sqrt {4 \cdot 1 - 3} + 2 \cdot 1 - 1 = 0$$ $$\sqrt 1 + 2 - 1 = 0$$ $$2 = 0$$

So, the original equation had no solutions .

Exercise 2: Solve equations

Equations with two radicals

Example 3: Solve $\sqrt {3x + 4} - \sqrt {2x + 1} = 1$

First thing to do is get one of the square roots by itself.

$$\sqrt {3x + 4} - \sqrt {2x + 1} = 1$$ $$\sqrt {3x + 4} = 1 + \sqrt {2x + 1} $$ $${(\sqrt {3x + 4} )^2} = {(1 + \sqrt {2x + 1} )^2}$$ $$3x + 4 = {1^2} + 2 \cdot 1 \cdot \sqrt {2x + 1} + {(\sqrt {2x + 1} )^2}$$ $$3x + 4 = 1 + 2\sqrt {2x + 1} + 2x + 1$$

We have managed to eliminate one of square roots!! We will continue to work this problem as we did in the previous examples.

$$3x + 4 = 1 + 2\sqrt {2x + 1} + 2x + 1$$ $$3x + 4 = 2 + 2x + 2\sqrt {2x + 1} $$ $$3x + 4 - 2 - 2x = 2\sqrt {2x + 1} $$ $${(x + 2)^2} = {(2\sqrt {2x + 1} )^2}$$ $${x^2} + 2 \cdot 2 \cdot x + {2^2} = {2^2}(2x + 1)$$ $${x^2} + 4x + 4 = 8x + 4$$ $${x^2} - 4x = 0$$ $$x(x - 4) = 0$$ $${x_1} = 0$$ $${x_2} = 4$$

Let's check both possible solutions. We will start with x = 0 .

$$\sqrt {3x + 4} - \sqrt {2x + 1} = 1$$ $$\sqrt {3 \cdot 0 + 4} - \sqrt {2 \cdot 0 + 1} = 1$$ $$\sqrt 4 - \sqrt 1 = 1$$ $$1 = 1 \Rightarrow OK$$

Now let's check x = 4 .

$$\sqrt {3x + 4} - \sqrt {2x + 1} = 1$$ $$\sqrt {3 \cdot 4 + 4} - \sqrt {2 \cdot 4 + 1} = 1$$ $$\sqrt {16} - \sqrt 9 = 1$$ $$1 = 1 \Rightarrow OK$$

Exercise 3: Solve equations

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A framework for solving parabolic partial differential equations

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Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and movies as well as the fabrication of complex geometric shapes using tools like 3D printing.

Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Among the many PDEs used in physics and computer graphics, a class called second-order parabolic PDEs explain how phenomena can become smooth over time. The most famous example in this class is the heat equation, which predicts how heat diffuses along a surface or in a volume over time.

Researchers in geometry processing have designed numerous algorithms to solve these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of these potentially nonlinear problems.  In a paper recently published in the Transactions on Graphics journal and presented at the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that can be solved with techniques graphics researchers already have in their software toolkit. This framework can help better analyze shapes and model complex dynamical processes.

“We provide a recipe: If you want to numerically solve a second-order parabolic PDE, you can follow a set of three steps,” says lead author Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For each of the steps in this approach, you’re solving a simpler problem using simpler tools from geometry processing, but at the end, you get a solution to the more challenging second-order parabolic PDE.” To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

First, their algorithm advances a solution forward in time by solving the heat equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it. 
This step can be completed easily with linear algebra.

Now, imagine that the parabolic PDE has additional nonlinear behaviors that are not described by the spread of heat. This is where the second step of the algorithm comes in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.  While generic HJ equations can be hard to solve, Mattos Da Silva and coauthors prove that their splitting method applied to many important PDEs yields an HJ equation that can be solved via convex optimization algorithms. Convex optimization is a standard tool for which researchers in geometry processing already have efficient and reliable software. In the final step, the algorithm advances a solution forward in time using the heat equation again to advance the more complex second-order parabolic PDE forward in time.


Among other applications, the framework could help simulate fire and flames more efficiently. “There’s a huge pipeline that creates a video with flames being simulated, but at the heart of it is a PDE solver,” says Mattos Da Silva. For these pipelines, an essential step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and can be solved using the researchers’ framework.

The team’s algorithm can also solve the diffusion equation in the logarithmic domain, where it becomes nonlinear. Senior author Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the result of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly in the logarithmic domain. This enabled a more stable way to, for example, find a geometric notion of average among distributions on surface meshes like a model of a koala. Even though their framework focuses on general, nonlinear problems, it can also be used to solve linear PDE. For instance, the method solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the same direction heat is spreading. In a straightforward application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.

The researchers note that this project is a starting point for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For example, they focused on static surfaces but would like to apply their work to moving ones, too. Moreover, their framework solves problems involving a single parabolic PDE, but the team would also like to tackle problems involving coupled parabolic PDE. These types of problems arise in biology and chemistry, where the equation describing the evolution of each agent in a mixture, for example, is linked to the others’ equations.

Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor at the University of Southern California’s Viterbi School of Engineering. Their work was supported, in part, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.

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A flexible solution to help artists improve animation

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