Higher June 2022 Paper 1R Q23

(a) Express \(2x^2 - 12x + 3\) in the form \(a(x + b)^2 + c\) where \(a\), \(b\) and \(c\) are integers. (3)

The curve C has equation \(y = 2(x + 4)^2 - 12(x + 4) + 3\)

The point \(M\) is the minimum point on C

(b) Find the coordinates of \(M\) (2)

Mark scheme (a)

Scheme	Marks
\(2(x^2 - 6x) + 3\) or \(2\left(x^2 - 6x + \dfrac{3}{2}\right)\)	M1
\(2\left[(x - 3)^2 - 9\right] + 3\) or \(2\left[(x - 3)^2 - 3^2 + \dfrac{3}{2}\right]\) oe	M1
\(2(x - 3)^2 - 15\)	A1
	(3)

M1: or for one of \(a\), \(b\) or \(c\) correct
OR expanding \(a(x^2 + 2bx + b^2) + c\)

M1: or for two of \(a\), \(b\) or \(c\) correct
OR \(-12 = 2ab\) or \(3 = ab^2 + c\)

A1: accept \(a = 2\), \(b = -3\), \(c = -15\)