x^2 + 6x + 5 = (x + 1)(x + □)

5. Two numbers with product 5 and sum 6: 1 and 5.

x^2 + 13x + k factorises as (x + 4)(x + m), so k = □.

36. 4 + m = 13, so m = 9; k is their product, 4 × 9 = 36.

Factorise x^2 − 10x − 24. What do you get? A. (x − 2)(x − 12) B. (x + 2)(x + 12) C. (x + 2)(x − 12) D. (x − 2)(x + 12) Choose the right answer

C. (x + 2)(x − 12). Product −24, sum −10: 2 and −12, so it is (x + 2)(x − 12).

Factorisation · Ages 13–14

The cross method: leading coefficient 1

Two numbers: product c, sum b.

How to do it

Method

Two numbers: product c, sum b

x² + 7x + 12 = (x + 3)(x + 4)

Another way

Draw the cross

3x + (−5x) = −2x

In full

(x + p)(x + q) = x² + (p + q)x + pq, so to factorise x² + bx + c, find two numbers that multiply to c and add to b. x² + 7x + 12: 3 and 4 multiply to 12 and add to 7, so x² + 7x + 12 = (x + 3)(x + 4). Look at the signs first: if c is positive, the two numbers have the same sign, b's; if c is negative, one is positive and one negative, and the larger in size takes b's sign. x² − 2x − 15: product −15, sum −2, so 3 and −5, giving (x + 3)(x − 5). Or draw the cross: x and x on the left, 3 and −5 on the right; across, 3x and −5x add up to −2x, so it is right. The answer is the same.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
x² + 6x + 5 = (x + 1)(x + )
Show the answer and working
Answer: 5
Two numbers with product 5 and sum 6: 1 and 5.
Example 2
x² + 13x + k factorises as (x + 4)(x + m), so k = .
Show the answer and working
Answer: 36
4 + m = 13, so m = 9; k is their product, 4 × 9 = 36.
Example 3
Factorise x² − 10x − 24. What do you get?
- A. (x − 2)(x − 12)
- B. (x + 2)(x + 12)
- C. (x + 2)(x − 12)
- D. (x − 2)(x + 12)
Choose the right answer
Show the answer and working
Answer: C. (x + 2)(x − 12)
Product −24, sum −10: 2 and −12, so it is (x + 2)(x − 12).

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.x² + 17x + 72 = (x + 9)(x + )
Answer:
2.Factorise x² + 14x + 48. What do you get?
- A. (x + 14)(x + 1)
- B. (x + 6)(x − 8)
- C. (x + 6)(x + 8)
- D. (x − 6)(x − 8)
Choose the right answer
Answer:
3.x² + 4x − 32 = (x + 8)(x − 4)
True or false?
Answer:
4.x² + 12x + k factorises as (x + 4)(x + m), so k = .
Answer:
5.x² + 5xy + 4y² = (x + y)(x + y)
Answer:

All the answers

1. 8
2. C. (x + 6)(x + 8)
3. True
4. 32
5. 4

Free worksheet: The cross method: leading coefficient 1Working out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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