## 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 ax^{2} + bx + c = 0 can also be represented as y = ax^{2} + 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 x^{2} = a,

Let us see some examples

**Example 1: **

Find the solutions of the equation x^{2} = 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 x^{2} = 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 x^{2} = 144.

**Solution: **

Step1: Given equation is = 144 … (1)

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

x^{2} = 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.

x^{2} = 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.

x^{2} = -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 𝒂𝒙^{𝟐}+𝒃=𝒄

How to solve an equation in the form of ax^{2}+b=c?

First write the equation in the form of x^{2}=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 4x^{2}+5 = 69 using square roots.

**Solution:**

Step1: Given quadratic equation 4x^{2 }+5 = 69 … (1)

Step2: Now write in the form x^{2 }= a,

we get 4x^{2} = 69−5

4x^{2} = 64

x^{2} = 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 x^{2}– 1= 24 using square roots.

**Solution:**

Step1: Given quadratic equation x^{2}– 1 = 24 … (1)

Step2: Now write in the form x^{2} = a,

we get x^{2} = 24+1

x^{2} = 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 3x^{2}−4 = 26 using square roots.

**Solution: **

Step1: Given quadratic equation 3x^{2}−4 = 26 … (1)

Step2: Now write in the form x^{2}= a,

we get 3x^{2} = 26+4

x^{2} = 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 3x^{2}+9 = 69 using square roots.

**Solution: **

Step1: Given quadratic equation 3x^{2}+9 = 69..… (1)

Step2: Now write in the form 3x^{2}= 60,

we get 3x^{2}= 60

3x^{2} = 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,

x^{2} = 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

- Find the solutions of the equation x
^{2}= 1. - Find the solutions of the equation x² = 45.
- Find the solutions of the equation x
^{2}= 16. - Find the solutions of the equation x² = 9.
- Find the solutions of the equation x
^{2 }= 81. - Find the solution of quadratic equation x
^{2}– 1 = 1 using square roots - Find the solution of quadratic equation x² + 1 = 1 using square roots
- Find the solution of quadratic equation 5x
^{2}– 1 = 24 using square roots - Find the solution of quadratic equation 6x
^{2}-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?

### Concept Map

### What have we learned

- Solving quadratic equations using square roots
- Solving quadratic equation in the form x
^{2}= a - Solving equation in the form ax
^{2}+b = c

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