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# Quadratic Equations Using Square Roots

## Key Concepts

• Solve quadratic equations using square roots
• Apply Square Root for Quadratic Equation
• Understand Positive and Negative square root consideration

### Introduction

A quadratic equation of the form ax2 + bx + c = 0 can also be represented as y = ax2 + bx + c.

## Solving Quadratic Equation using Square Roots

How can we solve quadratic equation using square roots?

We will write the given quadratic equation in square form by adding a constant, then we will perform the square root to find the variable value.

### Use of Square Roots in Solving Quadratic Equations

To understand the concept of square roots in quadratic equations and how to solve the equation in the form x2 = a,

Let us see some examples

Example 1:

Find the solutions of the equation x2 = 121.

Solution:

Step1: Given equation is x² = 121 … (1)

Step2: By seeing the equation we remember that 121 is square of 11.

x² = 121

x = ±√121

x = ±11

The solutions of the quadratic equation are x = +11 and x = -11.

Example 2:

Find the solutions of the equation x2 = 100.

Solution:

Step1: Given equation is z² = 100 … (1)

Step2: By seeing the equation we remember that 100 is the square of 10.

x² = 100

x = ±√100

x = ±10

The solutions of the quadratic equation are x = +10 and x = -10.

Example 3:

Find the solutions of the equation  x2 = 144.

Solution:

Step1: Given equation is = 144 … (1)

Step2: By seeing the equation we remember that 144 is square of 12.

x2 = 144

x = ±√144

x = ±12

The solutions of the quadratic equation are x = +12 and x = -12.

Example 4:

Find the solutions of the equation x² = 64.

Solution:

Step1: Given equation is x² = 64 … (1)

Step2: By seeing the equation we remember that 64 is square of 8.

x2 = 64

x = ±√64

x = ±8

The solutions of the quadratic equation are x = +8 and x = -8.

Example 5:

Find the solutions of the equation x² = -36.

Solution:

Step1: Given equation is x²=-36.… (1)

Step2: By seeing the equation we remember that 64 is square of 8.

x2 = -36

x = ±√-36

There is no real number that can be multiplied to get a negative number for which square root can be obtained.

### Solve Quadratic Equations of the form 𝒂𝒙𝟐+𝒃=𝒄

How to solve an equation in the form of ax2+b=c?

First write the equation in the form of x2=a, where a is a real number.

Take the square root on each side of the equation.

Then solve it.

Example 1:

Find the solution of quadratic equation 4x2+5 = 69 using square roots.

Solution:

Step1: Given quadratic equation 4x2 +5 = 69 … (1)

Step2: Now write in the form x2 = a,

we get 4x2 = 69−5

4x2 = 64

x2 = 16

x = ±√16

x = ±4

The solutions of the quadratic equation are x = +4 and x = -4

Example 2:

Find the solution of quadratic equation x2– 1= 24 using square roots.

Solution:

Step1: Given quadratic equation x2– 1 = 24 … (1)

Step2: Now write in the form x2 = a,

we get x2 = 24+1

x2 = 25

x = ±√25

x = ±5

The solutions of the quadratic equation are x = +5 and x = – 5

Example 3:

Find the solution of quadratic equation 3x2−4 = 26  using square roots.

Solution:

Step1: Given quadratic equation 3x2−4 = 26 … (1)

Step2: Now write in the form x2= a,

we get 3x2 = 26+4

x2 = 30

x = ±√10

x = ±10

The solutions of the quadratic equation are x = +√10 and x = −√10

Example 4:

Find the solution of quadratic equation 3x2+9 = 69  using square roots.

Solution:

Step1: Given quadratic equation 3x2+9 = 69..… (1)

Step2: Now write in the form 3x2= 60,

we get 3x2= 60

3x2 = 20

x = ±2√5

x = ±20

The solutions of the quadratic equation are x = +√25 and x = – 25

### Real Life Example

A ladder is leaned on a wall, the height on the wall is 13 m, the ladder is 14 m away from the wall, what is the length of the ladder?

Solution:

Let the length of the ladder be ‘x’ m

Now we get,

x2 = 13²+14²,  from the Pythagorean theorem.

Now

√x = ±13²+14²

= ±√169+196

= ±√365

As the length of the ladder cannot be negative

The length of the ladder =√365 ≈ 19.1 m

## Exercise

1. Find the solutions of the equation x2 = 1.
2. Find the solutions of the equation x² = 45.
3. Find the solutions of the equation x2 = 16.
4. Find the solutions of the equation x² = 9.
5. Find the solutions of the equation x2 = 81.
6. Find the solution of quadratic equation x2 – 1 = 1 using square roots
7. Find the solution of quadratic equation x² + 1 = 1 using square roots
8. Find the solution of quadratic equation 5x2 – 1 = 24 using square roots
9. Find the solution of quadratic equation 6x2-13 23 using square roots. 10. A ladder is leaned against a tree, the height on the wall is 3 m, the ladder is 4 m away from the tree, what is the length of the ladder?

### What have we learned

• Solving quadratic equations using square roots
• Solving quadratic equation in the form x2 = a
• Solving equation in the form ax2+b = c

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