The idea behind completing the square is to change an equation of the form ax
2
+ bx + x = 0 into an equation of the form
( Ax + B )
2
= 0 .
The reason for doing this is because we equations in the second form can be solved using a method we already know, namely, by using the square root property. In order to complete the square, you do the following steps:
1. Make sure the coefficient of the x 2 equation by the coefficient of x 2 .
terms is 1 by dividing all the terms in the ex.
ax
2
+ bx + c = 0
↓ ax 2 a x 2
+
+ bx a
↓
+ b a x + c a c a
=
0 a
= 0 divide each term in the equation by simplify a
2. Get the terms with the variable on one side of the equation and get the terms without any variable on the other side of the equation.
ex.
x 2 + x 2 + b a x + b a x +
↓ c a
↓
− x 2 + c a c a
= 0
= 0 b a x = − a c
− c a subtract − simplify c a from both sides
3. Compute it.
1
2 of the coefficient of the x terms, then take that number and square ex.
b a
↓ multiply by
1
2
· b a
↓ simplify b b
2 a
↓ square
2
2 a
↓ simplify b
2
4 a 2
1
2
1

4. Take number you computed in the previous step and add it to both sides of the equation.
ex.
x x
2
2 x
+
+
2 b x 2 + a x + b a x = − c a
↓ b
2
4 a 2
↓
= − c a
+ b
2
4 a 2 b a
+ x b a
+ x b 2
4 a 2
↓
= − 4 ac
4 a 2
+ b 2
4 a 2
=
+ b 2 − 4 ac a 2 b 2
4 a 2 add b 2
4 a 2 to both sides common denominator on right side simplify
5. Factor the left side into a perfect square.
ex.
x 2 + x b a x +
+ b
2 a
2 b 2
4 a 2
↓
=
= b 2 − 4 ac a 2 b
2 − 4 ac a 2 factor
6. Use the square root property to finish solving the equation.
ex.
x + x x
+
+ b
2 a b
2 a
2
= b
2 − 4 ac a 2
↓
= ± q b 2 − 4 ac a 2 x b
2 a
+
− x b
2 a b
2 a
=
↓
= ±
√ b 2 − 4 ac
2 a
↓
= ±
√ b 2 − 4 ac
2 a
↓
− b ±
√ b 2 − 4 ac
2 a
− b
2 a use square root property simplify subtract simplify b
2 a from both sides
2