**To solve a quadratic with completing the square method, you will add and deduct: ¼ (coefficient of x) 2**

## Example 1:

Solve the quadratic: *x2 + 8x + 4 = 0*

### Solution:

You cannot find two numbers whose product is 4 and sum is 8. So, we will use completing the square method to solve the given quadratic. Express the given quadratic in the form of the identity:

*a2 + 2ab + b2, **x2 + 8x + 4 = 0, **(x2 + 2 × x × 4 + 42) – 42 + 4 = 0 *so, add 42tox2 + 8xto complete the square.

### Important Tip:

To solve a quadratic with completing the square method, you will add and deduct: ¼ (coefficient of x) 2

Subtract 42 to keep the given quadratic unchanged. Now, the quadratic:

*x2 + 8x + 4 = 0 is (x2 + 2 × x × 4 + 42) – 42 + 4 = 0,*

*(x + 4)2 – 12 = 0, i.e.*

*(x + 4)2 – [√ (12)] 2 = 0, i.e.*

*(x + 4)2 = [√ (12)] 2, so,*

*x + 4 = ± √ (12), so,*

*x + 4 = √12, or x + 4 = – √12, so*

*x = 4 + √12 or x = 4 – √12*

## Example 2:

Solve the quadratic: *2×2 + x – 4= 0*

### Solution:

You cannot find two numbers whose product is: (2) × (- 4) i.e. – 8 and whose sum is +1. Use the completing square method.

**Use the tips:**

- Divide the quadratic with the leading coefficient (2, here)
- Add and subtract ¼ (x coefficient) 2

*½ (2×2 + x – 4) = 0, i.e. x2 + x/2 – 2 = 0*

*Now add and subtract ¼ (x coefficient) 2 i.e. ¼ (1/2)2 = ¼(1/4) = 1/16*

*The quadratic becomes: x2 + x/2 + 1/16 – 1/16 – 2 = 0*

**Write x/2 as 2 × x × ¼ to express in complete the square form:**

x2 + x/2 + 1/16 – 1/16 – 2 = 0,

x2 + 2 × x × ¼ + 1/16 = 1/16 + 2,

x2 + 2 × x × ¼ + (¼)2 = 33/16

(x + ¼)2 = 33/16,

(x + ¼) = ±√ (33/4)

x + ¼ = +√ (33/4) or x + ½ = – √ (33/4)

x = ¼ + √ (33/4) or x = ¼ – √ (33/4)

x = (1 + √ 33)/4 or x = (1- √ 33)/4