iGCSE Mathematics (0580) :E2.2 Manipulate directed numbers.iGCSE Style Questions Paper 2

Question

Factorise completely:

(a) $ 1 + x – y – xy $

(b) $ 2x^3 – 18xy^2 $

▶️ Answer/Explanation

Solution

Ans:

(a) $ (1 + x)(1 – y) $

1. Group terms: $ (1 + x) – y(1 + x) $

2. Factor out common $ (1 + x) $ term

(b) $ 2x(x + 3y)(x – 3y) $

1. Factor out 2x: $ 2x(x^2 – 9y^2) $

2. Recognize difference of squares: $ x^2 – 9y^2 = (x + 3y)(x – 3y) $

3. Combine factors

Question

(a) $x^2 + 8x + 10 = (x + p)^2 + q$

Find the value of $p$ and the value of $q$.

(b) Solve $x^2 + 8x + 10 = 30$.

▶️ Answer/Explanation

Solution

Ans (a): p = 4, q = -6

Complete the square: $(x^2 + 8x) + 10 = (x + 4)^2 – 16 + 10 = (x + 4)^2 – 6$.

Ans (b): x = -10 or x = 2

Using part (a): $(x + 4)^2 – 6 = 30$ → $(x + 4)^2 = 36$ → $x + 4 = ±6$.

So $x = -4 ±6$ giving solutions -10 and 2.

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