There are 3 basic methods to solve an **Quadratic Equation: –**

1. By Quadratic Formula

2. By Splitting the middle term

3. By completing the square

In this session ,we will discuss how we can solve an equation by Quadratic formula:)

## By Quadratic Formula

As we discussed this before in the concept “ Basics of quadratic equations”. The **Quadratic Equation Formula** is: –

(α, β) = [-b ± √(b^{2} – 4ac)]/2a or we can write it as [-b ± √D]/2a As: –

D = b^{2 }– 4ac

Now, Let us understand the application of quadratic formula by finding the roots of some **Quadratic Equation**:

**( 1 ). ****X ^{2} **

**+**

**2x**

**–**

**35**

**=**

**0**

According to the general quadratic form: ax^{2} + bx +c = 0

Here , a = 1 , b = 2, c = -35

You can follow the simple following steps to get your roots: –

**Step 1 =** Firstly, find the discriminant using b^{2} – 4ac (to put in the quadratic formula): –

D = b^{2} – 4ac

= (2)2 – 4 x 1 x -35

= 4 + 140 = 144

Our discriminant is 144.

**Step 2 =** Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -2 ± √144 / 2*1

= -2 + 12 /2 = -2-12/2

= 10/2 = -14/2

= 5 = -7

So, the final roots of the quadratic equation are 5 and -7.

**( 2 ). ****2x ^{2}-6x+3 = 0**

Here , a = 2 , b = -6, c = 3

**Step 1** = Firstly, find the discriminant using b^{2} – 4ac (to put in the quadratic formula): –

D = b2 – 4ac

= (-6)2 – 4 x 2 x 3

= 36 – 24 = 12

**Step 2 =** Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -(-6) ± √12 / 2x 2

= 6 + √12 /4 = 6 – √12 /4

So, the final roots of the **Quadratic Equation** are 6 + √12 /4 and 6 – √12 /4.

**( 3 ) . ****x ^{2}+5x+4 = 0**

Here , a = 1 , b = 5, c = 4

**Step 1** = Firstly, find the discriminant using b^{2} – 4ac (to put in the quadratic formula): –

D = b^{2} – 4ac

= (5)2 – 4 x 1 x 4

= 25 – 16 = 9

**Step 2 =** Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -5 ± √9 / 2 x 1

= -5 + 3 /2 = -5 – 3 / 2

= -2/2 = -8/2

= -1 = -4

So, the final roots of the quadratic equation are -1 and -4.

**( 4 ) . ****x ^{2} +10x + 9 = 0**

Here , a = 1 , b = 10, c = 9

Step 1 = Firstly, find the discriminant using b2 – 4ac (to put in the quadratic formula): –

D = b^{2} – 4ac

(10)^{2} – 4 x 1 x 9

= 100 – 36 = 64

**Step 2** = Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -10 ± √64 / 2 x 1

= -10 + 8 /2 = -10 – 8/2

= -2/2 = -18 / 2

= -1 = -9

So, the final roots of the quadratic equation are -1 and -9.

**( 5 ) . ****6x ^{2} -x – 5 = 0**

Here , a = 6 , b = -1, c = -5

**Step 1** = Firstly, find the discriminant using b2 – 4ac (to put in the quadratic formula): –

D = b^{2} – 4ac

= (-1)^{2} – 4 x 6 x -5

= 1 + 120 = 121

**Step 2 =** Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -(-1) ± √121 / 2 x 6

= 1 + 11 /12 = 1-11/12

= 12/12 = -10/12

= 1 = -5/6

So, the final roots of the quadratic equation are 1 and -5/6.

**6. ****12x ^{2} -25x = 0**

Here , a = 12 , b = -25, c = 0

**Step 1** = Firstly, find the discriminant using b2 – 4ac (to put in the quadratic formula): –

D = b^{2} – 4ac

= (-25)^{2} – 4 x 12 x 0

= 625-0 = 625

**Step 2 =** Now, put the discriminant in our quadratic formula.

x = [-b ± √(b^{2} – 4ac)]/2a

= -(-25) ± √625 / 2 x 12

= 25 + 25 /24 = 25-25/24

= 50/24 = 0/24

= 25/12 = 0

So, the final roots of the quadratic equation are 25/12 and 0.

I hope now, you can easily understand the topic “ solving a quadratic equation by quadratic formula”.

Read More : TRIANGLE: Definition, Parts, Properties,Types and Formulae

## Lets Practice

- 3x
^{2}+ x + 1 = 0 - x
^{2}+ 4x – 5 = 0 - 2x
^{2}+ x – 528 = 0 - x
^{2}-11x = -28 - x
^{2}– 4 = 0