Completing the Square (Edexcel IGCSE Further Pure Maths): Revision Note

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 ${\left(x+p\right)}^2-p^2$ where $p$ is half of $b$

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

  • Is $x^2+2x$ the same as ${\left(x+1\right)}^2-1^2$?

    • Yes: $\left(x+1\right)\left(x-1\right)-1^2=x^2+2x+1-1=x^2+2x$

• This works for negative values of $\mathit{b}$ too

  •  $x^2-20x$ can be written as ${\left(x-10\right)}^2-{\left(-10\right)}^2$ which is ${\left(x-10\right)}^2-100$

  • 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 ${\left(x+5\right)}^2-5^2$

  • add 9:  ${\left(x+5\right)}^2-5^2+9$

  • simplify the numbers:  ${\left(x+5\right)}^2-25+9$

  • answer: ${\left(x+5\right)}^2-16$

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

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

  • $a{x}^2+bx+c=a\left[x^2+\frac{b}{a}x\right]+c$

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

  • For example,  $4x^2+16x+5=4\left[x^2+4x\right]+5$

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

  • This gives $a\left[{\left(x+p\right)}^2-p^2\right]+c$

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

  • $4\left[x^2+4x\right]+5=4\left[{\left(x+2\right)}^2-4\right]+5$

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

  • $a{\left(x+p\right)}^2-a{p}^2+c$

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

    • Usually you are asked to give your final answer in the form  $a{\left(x+p\right)}^2+q$ 

    • Here  $q=-a{p}^2+c$

  • $4\left[{\left(x+2\right)}^2-4\right]+5=4{\left(x+2\right)}^2-16+5=4{\left(x+2\right)}^2-11$

• For quadratics like $\mathbf{-}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{x}\mathbf{+}\mathit{c}$, 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 $y={\left(x+p\right)}^2+q$ then the turning point is at $\left(-p,q\right)$

    • 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 $y=a{\left(x+p\right)}^2+q$ then the turning point is still at $\left(-p,q\right)$

    • 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