Solving Quadratic Equations by Completing the Square
The equation must be in standard form:
ax 2 + bx + c = 0
For example:
Solve the equation by completing the square:
2 x 2 + 12 x + 3 = 0
Therefore a = 2, b = 12, and c = 3
Step 1
If a = 1, then go on to Step 2. Otherwise, divide both sides of the equal sign by a.
Since a is not equal to 1. . .
3
2 x 2 12 x 3 0
2
Divide every term by 2. . . ö
+
+ =
. . . to get ö x + 6 x + = 0
2
2
2 2
2
Step 2
Get all variable terms on one side of the equation and all constants on the other side.
x 2 + 6x +
3
−3
= 0 ö x 2 + 6x =
2
2
Step 3
2
⎛b⎞
Complete the square for the resulting binomial by adding ⎜ ⎟ to both sides of the equation.
⎝2⎠
2
2
−3 ⎛6⎞
−3
⎛6⎞
ö x 2 + 6x + ⎜ ⎟ =
+⎜ ⎟
2
2 ⎝2⎠
⎝2⎠
−3
2
2
2
b. Divide the 6 by 2 in the parentheses ö x + 6 x + (3) =
+ (3)
2
−3
2
c. Square the resulting 3 and add it to both sides ö x + 6 x + 9 =
+9
2
a. b = 6
ö x 2 + 6x =
d. Convert 9 to a fraction to add constants on right side of equation ö x
e. Add the fractions on the right side of the equation ö
The Math Center
■
Valle Verde
■
x 2 + 6x + 9 =
Tutorial Support Services
2
+ 6x + 9 =
− 3 18
+
2
2
15
2
■
EPCC
1
Step 4
Factor the resulting perfect square trinomial and write it as the square of a binomial.
x 2 + 6x + 9 =
15
15
2
ö ( x + 3) =
2
2
Step 5
Use the square root property to solve for x.
a. Take the square root of both sides of the equation . . . ö
( x + 3)2
b. Subtract 3 from both sides to isolate the variable ö x = −3 ±
c. This leaves the final answers of ö
x = −3 +
15
15
. . . to get ö x + 3 = ±
2
2
=
15
2
15
15
and x = −3 −
2
2
Practice Exercises:
1.
3x 2 − 4 x = 4
2.
y 2 + 8 y + 18 = 0
3.
p2 + p − 2 = 0
4.
t 2 − 6t + 3 = 0
Answers:
1.
⎧ 2 ⎫
⎨− , 2 ⎬
⎩ 3 ⎭
2.
The Math Center
{− 4 + i
■
2 ,−4 − i 2
Valle Verde
■
}
3.
{− 2,1}
Tutorial Support Services
4.
■
EPCC
{3 +
6 ,3 − 6
}
2