Expanding double brackets

Foundation and Higher

The idea

Each term in the first bracket multiplies each term in the second: $(x + 3)(x - 5) = x^2 - 5x + 3x - 15$ , which simplifies to $x^2 - 2x - 15$ .

Watch out $(x + 4)^2$ is $x^2 + 8x + 16$ , not $x^2 + 16$ .

Worked exampleNon-calculator4 marks

Expand and simplify each expression.

(a) Expand and simplify $(x + 3)(x + 5)$

(b) Expand and simplify $(x - 4)(x + 2)$

(a) Multiply each term in the first bracket by each term in the second. $x^2 + 5x + 3x + 15$
Collect the $x$ terms. $x^2 + 8x + 15$
(b) $x^2 + 2x - 4x - 8 = x^2 - 2x - 8$

Answer (a) $x^2 + 8x + 15$ ; (b) $x^2 - 2x - 8$

Question 1Non-calculator3 marks

A rectangle has length $(x + 4)$ cm and width $(x + 2)$ cm.

(a) Write an expression for the area of the rectangle. Expand and simplify your answer.

(b) Work out the area when $x = 3$ .

Question 2Non-calculator4 marks

A square has sides of length $(x + 4)$ cm.

(a) Expand and simplify an expression for its area.

(b) Each side of the square is increased by 1 cm. Find an expression for the increase in area.

Question 3Non-calculator3 marks

Expand and simplify each product.

(a) Expand and simplify $(2x - 3)(x + 4)$ .

(b) Expand and simplify $(3 - x)(3 + x)$ .

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