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 x2 + 8x
to make a perfect square trinomial.

x2 is x2.

+8x is
+2 times
x times,
(+8x)/(+2⋅x), 4.

Write +42.

Write the same +42
on the right side.

So x2 + 8x = 5 becomes
x2 + 2⋅x⋅4 + 42 = 5 + 42.

x2 + 2⋅x⋅4 + 42 = (x + 4)2

Factor a Perfect Square Trinomial

+42 = +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 x2 - 7x
to make a perfect square trinomial.

x2 is x2.

-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 x2 - 7x = -11 becomes
x2 - 2⋅x⋅(7/2) + (7/2)2 = -11 + (7/2)2.

x2 - 2⋅x⋅(7/2) + (7/2)2
= (x - 7/2)2

+(7/2)2
= +72/22
= +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 = √22 = 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.