# Quadratic Equation: Completing the Square

How to solve a quadratic equation by completing the square: 2 examples and their solutions.

## Example 1

### Example

### Solution

Move the constant term -5 to the right side.

Use x^{2} + 8x

to make a perfect square trinomial.

x^{2} is x^{2}.

+8x is

+2 times

x times,

(+8x)/(+2⋅x), 4.

Write +4^{2}.

Write the same +4^{2}

on the right side.

So x^{2} + 8x = 5 becomes

x^{2} + 2⋅x⋅4 + 4^{2} = 5 + 4^{2}.

x^{2} + 2⋅x⋅4 + 4^{2} = (x + 4)^{2}

Factor a Perfect Square Trinomial

+4^{2} = +16

5 + 16 = 21

(x + 4)^{2} = 21

Square root both sides.

Then x + 4 = ±√21.

Quadratic Equation: Square Root

Move +4 to the right side.

Then x = -4 ± √21.

So x = -4 ± √21.

## Example 2

### Example

### Solution

Move the constant term +11 to the right side.

Use x^{2} - 7x

to make a perfect square trinomial.

x^{2} is x^{2}.

-7x is

-2 times

x times,

(-7x)/(-2⋅x), 7/2.

Write +(7/2)^{2}.

Write the same +(7/2)^{2}

on the right side.

So x^{2} - 7x = -11 becomes

x^{2} - 2⋅x⋅(7/2) + (7/2)^{2} = -11 + (7/2)^{2}.

x^{2} - 2⋅x⋅(7/2) + (7/2)^{2}

= (x - 7/2)^{2}

+(7/2)^{2}

= +7^{2}/2^{2}

= +49/4

Power of a Quotient

To solve -11 + 49/4,

change -11 to -44/4.

-44/4 + 49/4 = 5/4

(x - 7/2)^{2} = 5/4

Square root both sides.

Then x - 7/2 = ±√5/4.

±√5/4 = ±[√5/√4]

Divide Radicals

√4 = √2^{2} = 2

Square Root

x - 7/2 = ±[√5/2]

Move -7/2 to the right side.

Then x = 7/2 ± √5/2.

7/2 ± √5/2 = [7 ± √5]/2

So x = [7 ± √5]/2.