Section 8-6
Completing the Square
The idea is that: "(unknown stuff) squared equals a number" is an easy equation to solve, using square roots. So we want to put quadratic equations into that form.
Suppose you are given the quadratic equation
0 = x2 + bx + c.
If c = b2/4, then we already have a perfect square... all we need to do is take the square root. If not, the "completing the square trick" is to add something to c to get b2/4, and then subtract that something from the other side.
We'll give three progressively more difficult examples, starting with a simple square root problem and building up.
Example 1:
Solve for x in the equation:
x2 = 49
√(x2) = √49. . . . . . take square root of both sides
x = +7 or –7 . . . . . . there are two numbers whose square is 49
Example 2:
Solve for x in the equation:
(x + 6)2 = 5
x + 6 = √5 or –√5. . . . . . . .take square root of both sides
x = –6 + √5 or –6 –√5. . . . subtracting 6 from both sides
Example 3:
Solve x2 – 8x + 15 = 0 by completing the square.
x2– 8x + 15 = 0 . . . . . what perfect square starts with x2 – 8x? Half of -8 squared is (–4)2 = 16, so:
x2– 8x + 16 – 1 = 0
(x – 4)2 = 1 . . . take square root of both sides
x – 4 = 1 or – 1
x = 1 + 4 = 5, OR x = –1 + 4 = 3. . . add 4 to both sides
So the two solutions are 5 and 3.
