Quadratic Polynomials

Completing the Square

We now introduce the method of completing the square, which can be applied to solving any quadratic equation.

First we deal with the case:  x² + bx + c   (the leading coefficient is 1)

We want to write   x² + bx + c = (x + h)² − k   (known as completing the square)

Note    (x + a)² = x² + 2ax + a²

Rearranging gives    x² + 2ax = (x + a)² − a²

Write    b =2a    or   a =b/2

Hence,          

Example 9.

Study a few more examples.

x2 + x = (x + )2 − ()2 = (x + )2 − ()2

Exercise

Now try to solve a few on your own.

Write your answers in decimal form. x2 + x = (x + )2 − ()2

We can use this method to complete the square for any quadratic with a coeficient of 1 for x².

Examples

x² + 6x + 31 = (x + 3)² − (3)² + 31 = (x + 3)² + 22

x² − 11x − 3 = (x − 5.5)² − (5.5)² − 3 = (x − 5.5)² − 33.25

Exercise

Now try some of these exercises:

x2 x

Complete the square =  (x + )2 +

Completing the square allows us to find the roots of a quadratic equation.

Examples

Find the roots of   x² + 6x + 5:

x² + 6x + 5 = (x + 3)² − 3² + 55 = (x + 3)² − 4

The roots are given by   x² + 6x + 5 = (x + 3)² − 4 = 0

That is,   (x + 3)² = 4

Thus   x + 3 = 2   or   x + 3 = −2

That is,   x = 2 − 3 = −1   or   x = −2 − 3 = −5

Find the roots of   x² + 10x + 12 :

x² + 10x + 12 = (x + 5)² − (5)² + 12 5 = (x + 5)² − 13

The roots are given by   x² + 10x + 12 = (x + 5)² − 13 = 0

That is,   (x + 5)² = 13

Thus   x + 5 =    or   x + 5 = − and so the roots are − 5 and − − 5.

Exercise

Now try some of these exercises:

x2 x = 0

Complete the square =  (x + )2 −

Deduce the roots x =     or   x =

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