(i) Write \(x^2 + 4x - 16\) in the form \((x + a)^2 - b\). [3]
(ii) Solve the equation \(x^2 + 4x - 16 = 0\). Give your answers in surd form as simply as possible. [4]
(b) Sketch the graph of \(y = x^2 + 4x - 16\), showing clearly the coordinates of any turning points.
[3]
Mark scheme (a)
Answer
Marks
Part marks and guidance
(i) \((x + 2)^2\) final answer
1
– 20 final answer
2
FTtheir \((x + 2)^2\) final answer M1 for after \((x + a)^2\) shows \(-16 -\) their \(a^2\)
(ii) \(-2 \pm 2\sqrt{5}\)
4
B3FTtheir (a)(i) for \(-2 + \sqrt{20}\) or \(-2 - \sqrt{20}\) or better or M2 for \(x + 2 = (\pm)\sqrt{20}\) FT their (a)(i) or M1 for \((x + 2)^2 = 20\) FT their (a)(i) OR Alternative method B3 for \(\dfrac{-4 \pm 4\sqrt{5}}{2}\) or M2 for \(\dfrac{-4 \pm \sqrt{80}}{2}\) oe or M1 for \(\dfrac{-4 \pm \sqrt{4^2 - 4 \times 1 \times -16}}{2 \times 1}\) oe
FT dep on expression in form \((x + a)^2 - b\) in part (a)(i)
Condone 1 slip e.g. 1 sign error, 1² for 1, short fraction line, short root
Mark scheme (b)
Answer
Marks
Part marks and guidance
U shaped parabola
1
Be generous for the U shape condone broken line Values must be shown but could be marked on axes. Mark intention Accept turning point = (–2, –20) written in working provided no contradiction on sketch If point (–2, –20) only plotted on graph in 3rd quadrant and no sketch then award 2 marks
with turning point in 3rd quadrant indicated at (–2, –20) soi
2
B1FT for turning point at (\(k\), –20) or (–2, \(k\)) soi FT their (a)(i)
