Solving Radical Equations

Mathematics is the only subject which stays with us till our whole life. Whether you have to count money, watch time, press a television channel number, dial a contact number, score in an exam or so many daily activities are nothing without Mathematics.

Mathematics has many branches, and one is ‘solving radical equations.’ Some of you may have heard about it or know it by a different name.

This article will get to know more about ‘solving radical equations.’

Table of Content 

What is Radical Equation? 

How to Solve Radical Equation? 

Solving Radical Equations in Case of Odd Index 

What is Radical Equation?

Before knowing about the radical equations, first, understand what is radical. Radical is known as a symbol or a square root. It denotes the square root or any other root of a given expression.

Now, move to the radical equation. The radical equation is an equation in which at least one variable expression adheres inside a radical. Usually, this radical is the square root.

Therefore, the symbol

                                      nx

 indicates a root called radical or radical expression and is read as the nth root of x. At the same time, the equation containing a radical expression is called a radical equation.

Some examples of radical expression are given as follows:

 5, 4z, 17a+1, 664, 5±29, etc.

Do you know?

Radical equations play a significant role in Science, Mathematics, Engineering, and even  Music.

How to Solve Radical Equations?

In radical equations, a variable is under a radical. The term ‘solving radical equations’ means solving the radical equation and getting the variable’s value in the expression.

The general method for solving radical equations is to follow the given below steps:

1. Isolate the radical expression concerning the variable. If more than one radical expression involves the variable, isolate one of those.
2. Raise both sides of the equation to the power of the index of the radical.
3. Then eliminate the radical.
4. If there is still a radical equation, repeat steps 1, 2, and 3. Otherwise, solve the resulting equation and check its answer in the original equation.

11. 3 Solving radical equations containing an even index by raising the power of both sides to the index may introduce an algebraic solution that does not make the original equation true. Such solutions are known as extraneous solutions.

Let’s take a few examples to understand how to solve radical equations.

Example 1: Solve 5x-4 – 9 = 0.

Solution:

Step 1: Isolate the radical by adding 9 to both sides.

5x-4 – 9 + 9 = 0 + 9

5x-4 = 9

Step 2: Raising both sides of the index to the power of the index, i.e., 2.

[5x-4]²  = (9)²

Step 3: Solve the equation for the variable.

5x – 4 = 81

5x = 81 + 4

5x = 85

x = 85/5

x = 17

Hence, x = 17 for 5x-4 – 9 = 0.

Example 2: Solve 7 + a-3 = 1.

Solution:

Step 1: Isolate the radical by subtracting 7 on both sides.

7 + a – 3 – 7 = 1 – 7

a-3 = -6

Step 2: Raising both sides of the index to the power of the index, i.e., 2.

[a-3]² = (-6)²

Step 3: Solve the equation for the variable.

(a-3) = 36

a = 36 + 3

a = 39

Hence, a = 39 for 7 + a-3 = 1.

Example 3: Solve 2x+9 – 5 = 0.

Solution:

Step 1: Isolate the radical by adding 5 on both sides.

2x+9 – 5 + 5 = 0 + 5

2x+9 = 5

Step 2: Raising both sides of the index to the power of the index, i.e., 2.

[2x+9]² = (5)²

Step 3: Solve the equation for the variable.

(2x + 9) = 25

2x = 25 + 9

2x = 34

Divide both sides by 2, and we get,

x = 17

Hence, for x = 17, 2x+9 – 5 = 0.

Many more such examples can help you learn how to solve radical equations. You can also use solving radical equations calculators to get the answer to the given radical equations.

How to Solve Radical Equations Using Calculators?

Solving radical equations calculators are used to calculate the value of radical equations. An online tool helps calculate the variable’s value for the given radical equations. Solving radical equations calculators can solve the radical equations in a few seconds.

Suppose you want to use these solving radical equations calculators. In that case, you must enter the given radical equation in the input box.

How to use Solving Radical Equations Calculators?

To use such an online calculator, you need to follow solving radical equations calculators step by step. And these steps are given below:

Step 1: Go to the search bar and search online for solving radical equations calculators.

Step 2: Enter the radical equation in the given input box of the solving radical equations calculator.

Step 3: Click on the ‘Solve’ button to calculate the variable’s value for the given radical equation.

Step 4: Click on the ‘Reset’ button to clear all entered data and enter the new radical equation.

Solving Radical Equations in Case Of Odd Index

In the case of radicals with odd indexes, it is possible to have negative answers. Such problems also have solutions.

Steps to solve such problems are as follows:

Step 1: Isolate the radical expression.

Step 2: Raise both sides of the equation to the index of the radical. If there is a cubic index, then a cube on both sides.

Step 3: Solve to get the value of the variable.

Let’s take a look at how to solve radical equations with an odd index with the help of an example.

Example: Solve 32x+3 + 5 = 2.

Solution:

Step 1: Isolate the radical expression by subtracting 5 on both sides.

32x+3 + 5 – 5 = 2 – 5

32x+3 = -3

Step 2: Raising both sides of the index to the power of the index, i.e., 3.

[32x+3]³ = (-3)³

Step 3: Solve the equation for the variable.

2x + 3 = -27

2x = -27 – 3

2x = -30

Dividing both sides by 2, we get,

x = -15

Hence, x = -15 for  32x+3 + 5 = 2.

Solving Radical Equations in case of More than One Radical Expression

How do we solve radical equations if there is more than one radical expression? Don’t worry! It is the same as before. You need to repeat the steps one by one for each radical expression. It will take longer, but there is no difficulty with 11.3 radical solving equations.

Let’s understand it more clearly by taking an example.

Example: Solve 2x-5 – x-1 = 1.

Solution:

Step 1: Isolate the radical expressions.

2x-5 = 1 + x-1

Step 2: Raising both sides of the index to the power of the index, i.e., 2.

[2x-5]² = [1 + x-1]²

2x – 5 = (1)² + (x-1)² + 2*(1)*(x-1)

2x – 5 = 1 + (x – 1) + 2*x-1

Step 3: Solve for the value of the variable.

2x – 5 = x + 2*x-1

2x – 5 – x = 2*x-1

2*x-1 = x – 5

Step 4: Raising both sides of the index to the power of the index, i.e., 2.

[2*x-1]² = (x – 5)²

4*(x – 1) = x² + 25 – 10x

4x – 4 = x² + 25 – 10x

x² + 25 – 10x – 4x + 4 = 0

x² – 14x + 29 = 0

Step 5: Solving the quadratic equation.

                x = -bb²-4ac2a

Here, a = 1, b = -14, c = 29

Hence,

x = -(-14) (-14)² – 4(1)(29)2(1)

x = 14 (196) – 232

x = 14 802

x = 14 8.9442

x = 11.472, 2.528

Step 6: Check all the answers.

On checking the value of roots with the radical equation, we get x = 11.472 as the solution.

Such radical equations are examples of extraneous solutions.

Some More Examples Of Solving Radical Equations

Here are some more examples to understand how to solve radical equations.

Example 1:  Solve x . x-7 = 12.

Solution:

As in this problem, radical expressions are already isolated. Hence, we will move to the next step.

Step 1: Raising both sides of the index to the power of the index, i.e., 2.

[x . x-7 ]² = (12)²

x.(x – 7) = 144

x² – 7x = 144

x² – 7x – 144 = 0

Step 2: Solving the quadratic equation.

   x = -bb²-4ac2a

Here, a = 1, b = -7, and c = -144

Therefore,

x = -(-7) (-7)² – 4(1)(-144)2(1)

x = 7 (49) + 5762

x = 7 252

x = -9, 16

Step 3: Check all the answers.

On putting values of x in the radical equation, we get that x = 16 is the correct answer.

Example 2: Solve 17x-x²-5 = 7

Solution:

As in this problem, radical expressions are already isolated. Hence, we will move to the next step.

Step 1: Raising both sides of the index to the power of the index, i.e., 2.

[17x-x²-5]² = (7)²

17x – x²-5 = 49

17x – 49 = x²-5

Again raising both sides of the index to the power of the index, i.e., 2.

(17x – 49)² = [x²-5]²

289x² + 2401 – 1666x = x²-5

289x² + 2401 – 1666x – x²+5 = 0

287x² -1666x + 2406 = 0

Step 2: Solving the quadratic equation.

  x = -bb²-4ac2a

Here, a = 287, b = -1666, and c = 2406

  x = -(-1666) (-1666)²-4(287)(2406)2a

On solving it further, we get

x = 401144, 3

Step 3: Checking all answers.

On putting values of x in the given radical equation, we get that x = 3 is the correct answer.

Solving Radical Equations Worksheet

Here are some problems of solving radical equations given below. You can try to solve them.

Q. Solve 2x+9 – x+1 = x+4.

Q. Solve (x – 3) = 4x+9.

Q. Solve 4 + x+2 = x.

Q. Solve (x + 4) = x+10.

Q. Solve x+4 = 3x.

Frequently Asked Questions

1. How do you solve radicals step by step?

Solve Radicals Step by Step:

1. First, find the square root of both sides of your radical.
2. Next, move any negative(-) signs to the other side of the equals sign.
3. Then, multiply both sides by the conjugate of each term in your original expression and simplify if necessary.
4. Finally, take the square root of both sides and solve for x.

2. What are examples of radical equations?

Radical equations are equations that contain a radical sign, which looks like this: √.

Here are some examples of radical equations:

√x + 3 = 5

2√3x – 4 = 10

(6√3) + 2 = 10

3. How do you solve a square Radical?

Steps to solve a radical square problem:

Step 1: Find the perfect square root for each term.

Step 2: Get rid of all terms that are not perfect squares, like x2 or x3

Step 3: Take the sum of all the remaining terms. That’s your solution!

4. What are the three steps to solving a radical equation?

The steps to solving a radical equation are:

Factor the equation, if possible.

Use the rational root test to determine whether or not there is a rational root. If there is, use the quadratic formula to find it.

Use the remainder property of algebraic fractions to solve for any other roots that may exist.

5. How do you simplify radicals easily?

If you’re trying to simplify radicals, there’s a simple formula to follow. The first step is to find the root that’s being pulled out of the radical. Then, if it’s a perfect square, multiply it by itself and pull out the root. If it’s not a perfect square, divide the numerator by the denominator and pull out the root. Finally, if your answer is an irrational number (such as √2), “simplify” it by multiplying it by itself until you get something rational (like 2).