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Factoring Quadratics Area Model

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Factoring Quadratics Area Model

A design team is turning quadratic expressions into the side lengths of rectangles. Use factor pairs to find the dimensions. For each question, show enough working to explain your reasoning.

Read the model

Use the given terms in Questions 1–2 to answer each question. Then factorise the quadratic expression in Question 3.

1.Which factorisation matches x² + 5x + 6?

(x + 1)(x + 6)
(x + 2)(x + 3)
(x + 5)(x + 1)
(x + 2)(x + 4)

2.The two terms containing x are 2x and 3x. Explain how their coefficients combine to make the middle term.

3.The area of a rectangle is x² + 7x + 12. Find two positive integers whose product is 12 and sum is 7, then write the rectangle’s side lengths as factors.

Build the factors

For a monic quadratic x² + bx + c, look for two numbers that multiply to c and add to b. Check each factorisation by expanding.

4.Factorise x² + 4x + 3. State the two numbers you used and write the factorisation.

5.Factorise x² + 8x + 15, then verify your answer by expanding the brackets.

6.Complete the factorisation: x² − x − 12 = (x + )(x + ).

7.True or false? x² − 6x + 8 factorises as (x − 2)(x − 4). Expand the brackets to justify your answer.

TrueFalse

8.Match each quadratic to its factorisation.

x² + 9x + 20
x² + 6x + 8
x² − 7x + 10

(x + 2)(x + 4)
(x + 4)(x + 5)
(x − 5)(x − 2)

Put the factors to work

Connect factorised expressions to dimensions and solutions.

9.A rectangular garden has area x² + 7x + 12 square metres and length x + 4 metres. Find its width as an expression. Then calculate the length, width and area when x = 3.

10.Solve x² + 5x + 6 = 0 by factorising. Show how the factors give both solutions.

11.A student says x² + 2x − 15 = (x + 5)(x + 3). Identify the error and give the correct factorisation.