Massey logo
Home > College of Sciences > Institute of Fundamental Sciences >
Maths First > Online Maths Help > Algebra > Quadratic Polynomials > Completing the Square
SEARCH
MASSEY
MathsFirst logo College of Science Brandstrip
  Home  |  Study  |  Research  |  Extramural  |  Campuses  |  Colleges  |  About Massey  |  Library  |  Fees  |  Enrolment

 

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,           x^2 + b*x = (x + b/2)^2 - (b/2)^2.

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 = square root of 13   or   x + 5 = − square root of 13 and so the roots are square root of 13 − 5 and square root of 13 − 5.

Exercise

Now try some of these exercises:

x2 x = 0

Complete the square

=  (x + )2

Deduce the roots

x    or   x =

<< Roots | Quadratic Polynomials Index | The Quadratic Formula >>

 

   Contact Us | About Massey University | Sitemap | Disclaimer | Last updated: November 21, 2012     © Massey University 2003