Completing the square and turning points

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 3 marks
$y = x^2 + 6x + 2$
(a) Write $x^2 + 6x + 2$ in the form $(x + a)^2 + b$ . (2)
(b) Write down the coordinates of the turning point of $y = x^2 + 6x + 2$ . (1)
Question 2Non-calculator · 5 marks
Exactly 24 m of fencing forms three sides of a rectangle against a straight wall. Each side perpendicular to the wall has length x.
(a) Write the area in completed-square form. (3)
(b) Find the greatest possible area. (2)
Question 3Non-calculator · 4 marks
$E = x^2 - 8x + 20$
(a) By completing the square, show that $E > 0$ for all values of $x$ . (2)
(b) Find the greatest value of $\dfrac{1}{x^2 - 8x + 20}$ . (2)

Answers and marks

(a) $(x + 3)^2 - 7$

(b) $(-3, -7)$

(a) $-2(x-6)^{2}+72$

(b) $72$ m²

(a) $E = (x - 4)^2 + 4 \ge 4 > 0$

M1 Writing $(x - 4)^2 + 4$ .
C1 Stating that a square is never negative, so $E$ is at least 4 and so positive.

(b) $\frac{1}{4}$

P1 Recognising the fraction is greatest when $E$ is least, and finding the least value 4.
A1 The correct answer, $\frac{1}{4}$ .

Completing the square and turning points

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