Year 9 · Unit 8 Ready for Year 10 · Lesson 2 of 6

When a bracket has a minus sign in it

Add the two numbers for the x term and multiply them for the number term. A grid with signed cells shows why, and a value of x checks every line.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Work out (−3) × (−7).
    Answer21: a negative times a negative is positive.
  2. Simplify −4x + 9x.
    Answer5x: start at −4 and add 9.
  3. From lesson 1: expand (x + 2)(x + 9).
    Answerx² + 11x + 18: the cells are x², 2x, 9x and 18.
Words used in this lesson
sign
whether a number is positive or negative. For example: −6 has a negative sign.
product
the result of multiplying. For example: the product of −2 and 5 is −10.
x term
the part of an expansion with x but not x². For example: in x² − 7x + 10 the x term is −7x.
number term
the part with no letter. For example: in x² − 7x + 10 it is 10.
cancel
add to zero, so that nothing is left. For example: 6x − 6x = 0.

Kofi or Lena?

Kofi and Lena both expand (x − 4)(x + 6). Kofi writes x² + 2x − 24. Lena writes x² − 2x + 24.

Predict first: who is right, or is neither? Before you decide, work out (x − 4)(x + 6) at x = 5 in your head, then each of their answers at x = 5. Then use the grid to see where each sign comes from.

2
−5
4
x² − 3x − 10the collected expansion
−3a + b, the number in front of x
−10a × b, the number term

(x + 2)(x − 5): the x terms are 2x − 5x = −3x and the number term is 2 × (−5) = −10, so x² − 3x − 10. At x = 4 both give −6.

Show me the steps
  1. a = 2, b = −5: the cells are x², 2x, −5x and −10. The x terms make 2x − 5x = −3x, so (x + 2)(x − 5) = x² − 3x − 10. At x = 4: 6 × (−1) = −6, and 16 + 8 − 20 − 10 = −6.
  2. a = −3, b = −4: the cells are x², −3x, −4x and +12, because (−3) × (−4) = 12. So (x − 3)(x − 4) = x² − 7x + 12.
  3. a = 6, b = −6: the x terms are 6x − 6x = 0, so they cancel and (x + 6)(x − 6) = x² − 36.
  4. Check a line with a value: (x − 3)(x − 4) at x = 1 is (−2) × (−3) = 6, and 1 − 7 + 12 = 6.

(x + a)(x + b) = x² + (a + b)x + ab. Add the two numbers for the x term; multiply them for the number term. Each number keeps its sign: x − 4 is x + (−4), so the number is −4.

Why: the two middle cells of the grid are ax and bx, which collect to (a + b)x; the corner cell is a × b. For Kofi and Lena the numbers are −4 and 6: the x term is (−4 + 6)x = 2x and the number term is (−4) × 6 = −24. Kofi is right. At x = 5 the brackets give 1 × 11 = 11, Kofi's line gives 25 + 10 − 24 = 11 and Lena's gives 39.

A common mistake: dropping a sign in the grid and hoping to put it back at the end. In (x − 8)(x + 1) the cells are x², −8x, x and −8: every cell made with −8 is negative. Write each sign in its cell as you go.

Worked examples

Expand (x − 7)(x + 3).

  1. The numbers are −7 and 3.
  2. x term: −7 + 3 = −4. Number term: (−7) × 3 = −21.
  3. x² − 4x − 21. Check at x = 1: (−6) × 4 = −24 and 1 − 4 − 21 = −24.

Expand (x − 2)(x − 6).

  1. The numbers are −2 and −6.
  2. x term: −2 + (−6) = −8. Number term: (−2) × (−6) = +12.
  3. x² − 8x + 12. Check at x = 3: 1 × (−3) = −3 and 9 − 24 + 12 = −3.

Your go. The first step is given. Expand (x + 8)(x − 3).

  1. The numbers are 8 and −3.
  2. Now find the x term and the number term yourself, and check with a value.
Check the answer

8 + (−3) = 5 and 8 × (−3) = −24: x² + 5x − 24. Check at x = 2: 10 × (−1) = −10 and 4 + 10 − 24 = −10.

Check before you go on

1. Which is (x − 3)(x − 8) written without brackets?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. (x + 7)(x − 2) = x² + px + q. What is q?

3. Expand (x − 5)(x − 4). What number is in front of x?

Level 2 · Building

4. Find the value of (x − 4)(x + 9) when x = −2.

5. (x + k)(x − 3) = x² + 4x − 21. What is k?

Level 3 · Stretch

6. (4x − 6)(x + b) has no x term at all. What is b? Give it as a decimal.

7. In the expansion of (x + a)(x − 8) the x term is −3x. What is the number term?

Something to think about

Find every pair of whole numbers a and b for which (x + a)(x + b) has the number term −12. What different x terms can you make?

One way to think about it

a × b = −12 with one number negative: 1 and −12, −1 and 12, 2 and −6, −2 and 6, 3 and −4, −3 and 4. Their sums give the x terms −11x, 11x, −4x, 4x, −x and x. Working backwards like this, from the expansion to the brackets, is called factorizing, and Year 10 starts there.

Exit ticket

8. What is the x term of (2x − 5)(x + 3)?