Completing the Square to get a Quadratic Function
into Standard ( h, k) Form ;   y = a(x - h)² +k

Example 1 Coefficient of Quadratic term equals 1

f(x) = x² -12x +5

Take half the coefficient of the "middle term" ( here , -12) and square it.

Add and subtract it as shown.

f(x) = x² - 12x + 36 + 5 - 36

Group as shown:

f(x) = (x² - 12x + 36 )  + 5 - 36

Factor and simplify

f(x) = (x - 6)² - 31

Example 2   Coefficient of Quadratic term not equal to 1

f(x) = 2x² - 6x + 5

First factor the coefficient of the squared term form the terms with the variable

f(x) = 2(x² - 3x) + 5

Now follow the rules above.

Take 1/2 of 3 , square it, add it and subtract it, being careful with the distributive law.

f(x) = 2(x² - 3x + 9/4) + 5 - 2(9/4)

Note: since you added 9/4 inside the parentheses , you were actually adding 2 times 9/4 and have to compensate for this when subtracting.

Factor and simplify

f(x) = 2(x - 3/2)² + 1/2

To double check your results graph the equation in both forms on the same graph using y1 ans y2. The graphs  should be identical.

© Joan Bookbinder 1998 1999 2001