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 𝒂𝒙^𝟐+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?

Concept Map

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