June 2016 Paper 1 Q12
12 You are given that \(\qquad x^2 + 6x + 2 \equiv (x + h)^2 + k\)
(a) Work out the values of \(h\) and \(k\). [2 marks]
(b) Write down the coordinates of the minimum point on the curve \(\quad y = x^2 + 6x + 2\) [1 mark]
(c) Solve the equation \(\qquad x^2 + 6x + 2 = 0\)
Give your answers in the form \(\ a \pm \sqrt{b}\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \((x + 3)^2 - 9\ (+ 2)\) | M1 | |
| \(h = 3\) and \(k = -7\) | A1 | |
| Alternative method 2 | ||
| \(x^2 + 2hx + h^2\ (+ k)\) or \(2hx = 6x\) or \(2h = 6\) or \(h^2 + k = 2\) | M1 | |
| \(h = 3\) and \(k = -7\) | A1 | |
Additional guidance
\(h = 3\) implies M1
| Answer | Mark | Comments |
|---|---|---|
| \((-3, -7)\) | B1 ft | ft their \(h\) and \(k\) from part (a) only if \(h \neq 0\) and \(k \neq 0\) |
Additional guidance
for their \(h\) and \(k\), the minimum point is \((-h, k)\)
| Answer | Mark | Comments |
|---|---|---|
| \(-3 \pm \sqrt{7}\) | B1 ft | ft their \(h\) and \(k\) from part (a) only if \(h \neq 0\) and \(k \neq 0\) |
Additional guidance
For their \(h\) and \(k\), the solutions are \(-h \pm \sqrt{(-k)}\)
If their \(k\) is \(\gt 0\) then \(\sqrt{(-k)}\) will be \(\sqrt{}\) of a negative number ... condone
Any use of the quadratic formula must be completely correct