Solving Quadratic Equations by Completing the Square

We saw last section that if we have a quadratic equation in the form
(something) ² = # you can use the square root property . In this section we will
learn a method to take all quadratic equations and write them in that form.
This method is called completing the square.

Before we can solve a quadratic equation by completing the square, we
must see what it takes to be a perfect square trinomial.

EXAMPLE: Which of the fol lowing are perfect square trinomials ?

CONCLUSION: When a trinomial is of the form x² + bx + c , it is a perfect
square if . In this case we get“a perfect
square”.

EXAMPLE De termine the constant that should be added to make each
expression a perfect square. trinomial:

How to construct a perfect square trinomial:
i.) Divide the coefficient of x by 2
ii.) Square this value ; this is the constant term on the perfect square
trinomial.

To apply this to solving quadratic equations we remember that what we add
to one side we add to the other side.

EXAMPLE: Solve x² − 8x + 4 = 0 by completing the square.
We know that we complete the square on the x² − 8x terms. So isolate
these terms.

Now complete the square:

This gives

Now use the square root method to solve:

Simplify the radical :

Now solve for x .

EXAMPLE: Solve 2x² = 3 − 5x by completing the square.
Isolate the x and x² terms.

We need the coefficient of x ² to be 1. So divide both sides by 2.

Now complete the square:

This gives

Now use the square root method to solve:

Simplify the radical:

Now solve for x .

Un like the first example, here we got “ nice solutions ”. This means that we
could have solved this one by factoring .

EXAMPLE Solve each quadratic equation by completing the square.