2x^2 + 21x + 49 = (2x + 7)(x + □)

7. 7 × □ = 49, so □ = 7. Check across: 2x × 7 + x × 7 = 21x.

4x^2 − 43x + 63 ≡ (4x + h)(x + k), so h = □ and k = □. Fill in each blank

−7, −9. hk = 63 and, across, 4k + h = −43: h = −7 and k = −9 fit both.

4x^2 + 43x + 63 = (4x + 7)(x + 9) True or false?

True. Check across: 4x × 9 + x × 7 must be 43x. The right one is (4x + 7)(x + 9).

Factorisation · Ages 13–14

The cross method: other leading coefficients

Split a and c; the cross products must add to b.

How to do it

Method

Split a and c; the cross products must add to b

2x² + 7x + 3 = (2x + 1)(x + 3)

Another way

Split the middle term

2x² + 6x + x + 3 = 2x(x + 3) + (x + 3)

In full

When x²'s coefficient isn't 1, split a and c each into two factors, set them in a cross, and multiply across and add: it must make b. 2x² + 7x + 3: 2 = 2 × 1 and 3 = 1 × 3; with 2x and x on the left and 1 and 3 on the right, across is 2x × 3 = 6x and x × 1 = x, and 6x + x = 7x. Right, so it is (2x + 1)(x + 3), each bracket a row of the cross. If not, swap the right-hand pair or split another way. Take out any common factor first: 4x² + 14x + 6 = 2(2x² + 7x + 3). Or split the middle term: a × c = 6, and 6 and 1 multiply to 6 and add to 7, so 2x² + 6x + x + 3 = 2x(x + 3) + (x + 3) = (2x + 1)(x + 3). The answer is the same.

Worked examples

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

Example 1
2x² + 21x + 49 = (2x + 7)(x + )
Show the answer and working
Answer: 7
7 × □ = 49, so □ = 7. Check across: 2x × 7 + x × 7 = 21x.
Example 2
4x² − 43x + 63 ≡ (4x + h)(x + k), so h = and k = .
Fill in each blank
Show the answer and working
Answer: −7, −9
hk = 63 and, across, 4k + h = −43: h = −7 and k = −9 fit both.
Example 3
4x² + 43x + 63 = (4x + 7)(x + 9)
True or false?
Show the answer and working
Answer: True
Check across: 4x × 9 + x × 7 must be 43x. The right one is (4x + 7)(x + 9).

Practice questions to try

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

1.3x² + 19x + 20 = (3x + 4)(x + )
Answer:
2.Factorise 3x² + 23x + 40. What do you get?
- A. (3x + 5)(x + 8)
- B. (3x − 8)(x − 5)
- C. (3x + 40)(x + 1)
- D. (3x + 8)(x + 5)
Choose the right answer
Answer:
3.4x² + 41x + 45 = (4x + 5)(x + 9)
True or false?
Answer:
4.Splitting the middle term of 2x² + 15x + 18: 2 × 18 = ; split 15x into x + x (the larger first), so that the two coefficients multiply to 2 × 18.
Fill in each blank
Answer:
5.Factorise 6x² + 34x + 20 completely. What do you get?
- A. 2(3x − 2)(x − 5)
- B. 2(3x + 5)(x + 2)
- C. 2(3x + 2)(x + 5)
- D. (6x + 4)(x + 5)
Choose the right answer
Answer:

All the answers

1. 5
2. D. (3x + 8)(x + 5)
3. True
4. 36, 12, 3
5. C. 2(3x + 2)(x + 5)

Free worksheet: The cross method: other leading coefficientsWorking 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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