WMA11/01 Oct 2019, Question 10: Factorising Quadratic Expressions

Part (a) · 2 marks

State the x values satisfying f(x) ≤ 0, using the sketch and factorised form.

1. 1f(x) = 0 at x = −5/2 (crosses) and x = 3 (touches, since (x − 3)² ≥ 0)
2. 2(x − 3)² ≥ 0, so f(x) ≤ 0 needs 2x + 5 ≤ 0 or x = 3
3. 3x ≤ −5/2 or x = 3

Answerx ≤ −5/2 or x = 3

Part (b) · 3 marks

Expand f(x) into the form ax³ + bx² + cx + d with integer coefficients a, b, c, d.

1. 1(x − 3)² = x² − 6x + 9
2. 2(2x + 5)(x² − 6x + 9) = 2x³ − 12x² + 18x + 5x² − 30x + 45
3. 3= 2x³ − 7x² − 12x + 45

Answerf(x) = 2x³ − 7x² − 12x + 45

Part (ci) · 1 mark

Give the coordinates of the point P where the curve meets the y-axis.

1. 1At x = 0, f(0) = 45
2. 2P = (0, 45)

AnswerP = (0, 45)

Part (cii) · 1 mark

Find the gradient of the curve at P.

1. 1f′(x) = 6x² − 14x − 12
2. 2At x = 0, gradient = −12

Answer−12

Part (di) · 2 marks

Write g(x), the curve translated 2 units in the positive x direction, in simplified factorised form.

1. 1Translation 2 units right: g(x) = f(x − 2)
2. 2g(x) = (2(x − 2) + 5)((x − 2) − 3)²
3. 3g(x) = (2x + 1)(x − 5)²

Answerg(x) = (2x + 1)(x − 5)²

Part (dii) · 1 mark

State the y intercept of y = g(x).

1. 1g(0) = (1)(−5)²
2. 2y-intercept = 25

Answer25 (the point (0, 25))

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