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Completing the Square (Cambridge (CIE) O Level Additional Maths): Revision Note

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Completing the Square

What is completing the square?

  • Completing the square is another way of writing a quadratic function

  • It means rewriting y space equals a x squared plus b x plus c in the form y space equals space a left parenthesis x plus p right parenthesis squared plus q

    • The key point is that x now only occurs once in the equation

  • It can be used to solve quadratic equations, sketch their graphs and to find the coordinates of the turning point

How do I complete the square?

The method used will depend on the value of the coefficient of the bold italic x to the power of bold 2 term in y space equals a x squared plus b x plus c

  • When bold italic a bold equals bold 1

    • p is half of b

    • q is c minus p squared

Example of completing the square

 

  • When a ≠ 1 

    • First take a factor of a out of the bold italic x to the power of bold 2 and bold italic x terms

    • Then continue as above

Harder example of completing the square

Examiner Tips and Tricks

  • Sometimes a question will explicitly use the phrase complete the square

  • Sometimes a question will use the form a open parentheses x plus p close parentheses squared plus q without using the phrase completing the square

Worked Example

Write 3 x squared minus 12 x plus k in the form a open parentheses x plus p close parentheses squared plus k plus q, where a comma space p and q are constants to be found, and k is an unknown constant.

The form required is 'completing the square' (do not be put off by the k, it is just a constant!).

STEP 1 - Take a factor of 3 out of the x squared and x terms - leaving k avoids awkward fractions.

3 x squared minus 12 x plus k equals 3 open parentheses x squared minus 4 x close parentheses plus k

STEPS 2 and 3 - Complete the square on the open parentheses x squared minus 4 x close parentheses part only. "p equals negative 4 over 2 equals negative 2" and "q equals 0 minus open parentheses negative 2 close parentheses squared equals 4".

3 open square brackets open parentheses x minus 2 close parentheses squared minus 4 close square brackets plus k

STEP 4 - Expand and simplify.

3 open parentheses x minus 2 close parentheses squared minus 12 plus k

i.e. a equals 3 comma space p equals negative 2 comma space q equals negative 12

Solving by Completing the Square

How do I solve a quadratic equation by completing the square?

  • To solve x2 + bx + c = 0 

    • replace the first two terms, x2 + bx, with (x + p)2 - p2 where p is half of b

    • this is called completing the square

      • x2 + bx + c = 0 becomes

        • (x + p)2 - p2 + c = 0 where p is half of b

    • rearrange this equation to make x the subject (using ±√)

  • For example, solve x2 + 10x + 9 = 0 by completing the square

    • x2 + 10x becomes (x + 5)2 - 52

    • so x2 + 10x + 9 = 0 becomes (x + 5)2 - 52 + 9 = 0

    • make x the subject (using ±√)

      • (x + 5)2 - 25 + 9 = 0

      • (x + 5)2 = 16

      • x + 5 = ±√16

      • x  = ±4 - 5

      • x  = -1 or x  = -9

  • If the equation is ax2 + bx + c = 0 with a number in front of x2, then divide both sides by a first, before completing the square 

Examiner Tips and Tricks

  • When making x the subject to find the solutions at the end, don't expand the squared brackets back out again!

    •  Remember to use ±√ to get two solutions

Worked Example

Solve 2 x squared minus 8 x minus 24 equals 0 by completing the square.

Divide both sides by 2 to make the quadratic start with x2   

x squared minus 4 x minus 12 equals 0  

Halve the middle number, -4, to get -2
Replace the first two terms, x2 - 4x, with (x - 2)2 - (-2)2  

open parentheses x minus 2 close parentheses squared minus open parentheses negative 2 close parentheses squared minus 12 equals 0  

Simplify the numbers  

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses x minus 2 close parentheses squared minus 4 minus 12 end cell equals 0 row cell open parentheses x minus 2 close parentheses squared minus 16 end cell equals 0 end table  

Add 16 to both sides  

open parentheses x minus 2 close parentheses squared equals 16  

Square root both sides
Include the ± sign to get two solutions  

x minus 2 equals plus-or-minus square root of 16 equals plus-or-minus 4  

Add 2 to both sides  

x equals plus-or-minus 4 plus 2  

Work out each solution separately

x = 6 or x = -2

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