Completing the Square (Edexcel IGCSE Further Maths)

Spec point — 2A

Completing the Square

How can I rewrite the first two terms of a quadratic expression as the difference of two squares?

• Look at the quadratic expression $x^{2}+bx+c$* *
• The first two terms can be written as the difference of two squares using the following rule

$x^{2}+bx$ is the same as (x+p)²−p² where $p$ is half of $b$

• Check this is true by expanding the right-hand side

  • Is $x^{2}+2x$ the same as (x+1)²−1²?

    • Yes: (x+1)(x−1)−1²=x²+2x+1−1=x²+2x

• This works for negative values of$b$ too

  • $x^{2}-20x$ can be written as (x−10)²−(−10)² which is (x−10)²−100
  • A negative $b$ does not change the sign at the end

How do I complete the square?

• Completing the square is a way to rewrite a quadratic expression in a form containing a squared bracket

• To complete the square on $x^{2}+10x+9$

  • Use the rule above to replace the first two terms, $x^{2}+10x$, with (x+5)²−5²
  • add 9: (x+5)²−5²+9
  • simplify the numbers: (x+5)²−25+9
  • answer: (x+5)²−16

How do I complete the square when there is a coefficient in front of the x² term?

• You first need to take$a$out as a factor of the $x^{2}$ and $x$ terms only

  • $a x²+b x+c=a [ x²+b / a x ] +c$

    • Use square-shaped brackets here to avoid confusion with round brackets later

  • For example, $4x²+16x+5=4 [ x²+4x ] +5$

• Then complete the square on the bit inside the square brackets: $x^{2}+\frac{b}{a}x$

  • This gives $a [ (x+p)²−p² ] +c$

    • where p is half of $\frac{b}{a}$

  • $4 [ x²+4x ] +5=4 [ (x+2)²−4 ] +5$

• Finally multiply this expression by the $a$ outside the square brackets and add the $c$

  • a (x+p)²−a p²+c

  • This looks far more complicated than it is in practice!

    • Usually you are asked to give your final answer in the form a (x+p)²+q
    • Here $q=-ap^{2}+c$

  • $4 [ (x+2)²−4 ] +5=4 (x+2)²−16+5=4 (x+2)²−11$

• For quadratics like $-x^{2}+bx+c$, do the above with $a=-1$

How do I find the turning point by completing the square?

• Completing the square helps us find the turning point on a quadratic graph

  • If y=(x+p)²+q then the turning point is at (−p comma q)

    • Notice the negative sign in the $x$-coordinate
    • This links to transformations of graphs (translating $y=x^{2}$ by $p$ to the left and $q$* *up)

  • If y=a (x+p)²+q then the turning point is still at (−p comma q)

    • It's a minimum point if $a>0$
    • It's a maximum point if $a<0$

• It can also help you create the equation of a quadratic when given the turning point

• It can also be used to prove and/or show results using the fact that any "squared term", i.e. the bracket (x ± p)² , will always be greater than or equal to 0

  • You cannot square a number and get a negative value

Exam Hint

• Expand your answer to check that you have completed the square correctly.

Worked Example (a) By completing the square, find the coordinates of the turning point on the graph of $y=x^{2}+6x-11$.

Find half of $+6$ (call this $p$)

$p=\frac{6}{2}=3$

Write $x^{2}+6x$ in the form (x+p)²−p²* *

$x^{2}+6x$ is the same as (x+3)²−3²

• *

Put this result into the equation of the curve

y=(x+3)²−3²−11

Simplify the numbers

y=(x+3)²−20

Use the fact that the turning point of y=(x+p)²+q is at (−p comma q)

Here $p=3$ and $q=-20$

Final answer: Turning point at$(-3,-20)$

(b) Write $-3x^{2}+12x+24$ in the form a (x+p)²+q

Factorise $-3$ out of the first two terms only Use square-shaped brackets

$−3 [ x²−4x ] +24$

Complete the square on the $x^{2}-4x$ inside the brackets (write in the form (x+p)²−p² where $p$ is half of $-4$)

$−3 [ (x−2)²−(−2)² ] +24$

Simplify the numbers inside the brackets (−2)² is $4$

$−3 [ (x−2)²−4 ] +24$

Multiply all the terms inside the square-shaped brackets by $-3$

−3 (x−2)²+12+24

Simplify the numbers

−3 (x−2)²+36

This is now in the form a (x+p)²+q where $a=-3$, $p=-2$ and $q=36$

$-3(x-2)^{2}+36$