Bradley's Maths
Completing the Square to Find the Turning Point
GCSE all examination boards and IGCSE Cambridge (0580)
Exam Question
Consider the expression $-5-6x-x^2$.
(a) Write the expression in the form $q-(x+p)^2$.
(b) Hence write down the coordinates of the turning point of $y=-5-6x-x^2$.
Solution
First rewrite the quadratic:
$$y=-x^2-6x-5$$Factor out $-1$:
$$y=-(x^2+6x)-5$$Complete the square:
$$x^2+6x=(x+3)^2-9$$Substitute:
$$y=-[(x+3)^2-9]-5$$Simplify:
$$y=4-(x+3)^2$$The turning point is $(-3,4)$.
The Head Teacher's Eye: Get the Turning Point Right!
If the completed square format is $(x+p)^2+q$ then the $x$-coordinate of the turning point is the number which will make $(x+p)=0$ that is $-p$ and the $y$-coordinate is $q$. If the bracket is $(x-p)$ then the $x$-coordinate becomes $p$.
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Topics covered in this question
- Completing the Square when the coefficient of $x^2$ is negative
- Using the completed square form of a quadratic to write down the coordinates of the turning point