Completing the Square

The technique of completing the square is presented here primarily to justify the quadratic formula, which will be presented next. However, the technique does have applications besides being used to derive the quadratic formula. In analytic geometry, for example, completing the square is used to put the equations of conic sections into standard form.

Before considering the technique of completing the square, we must define a perfect square trinomial.

Perfect Square Trinomial

What happens when you square a binomial?

·        Note that the coefficient of the middle term (2a) is twice the square root of the constant term (a2)

·        Thus the constant term is the square of half the coefficient of x

·        Important: These observations only hold true if the coefficient of x is 1.

This means that any trinomial that satisfies this condition is a perfect square. For example,

            x2 + 8x + 16

is a perfect square, because half the coefficient of x (which in this case is 4) happens to be the square root of the constant term (16). That means that

x2 + 8x + 16 = (x + 4)2

Multiply out the binomial (x + 4) times itself and you will see that this works.

The technique of completing the square is to take a trinomial that is not a perfect square, and make it into one by inserting the correct constant term (which is the square of half the coefficient of x). Of course, inserting a new constant term has to be done in an algebraically legal manner, which means that the same thing needs to be done to both sides of the equation. This is best demonstrated with an example.

Example:   

Given Equation:

Move original constant to other side:

Add new constant to both sides
(the square of half the coefficient of x):

Write left side as perfect square:

Square root both sides
(remember to use plus-or-minus):

Solve for x:

Notes

·        Finds all real roots. Factoring can only find integer or rational roots.

·        When you write it as a binomial squared, the constant in the binomial will be half of the coefficient of x.

If the Coefficient of x2 is Not 1

First divide through by the coefficient, then proceed with completing the square.

Example:  

Given Equation:

Divide through by coefficient of x2:
(in this case a 2)

Move constant to other side:

Add new constant term:
(the square of half the coefficient of x, in this case 9/16):

Write as a binomial squared:
(the constant in the binomial is half the coefficient of x)

Square root both sides:
(remember to use plus-or-minus)

Solve for x:

                 Thus               x = ½  or  x = -2