Year 9 · Unit 8 Ready for Year 10 · Lesson 4 of 6

Simplify, then prove yourself wrong (or not)

Expand every product, bracket it, collect like terms, then check by putting a number into the first line and the last. One value that disagrees proves a mistake; one value that agrees proves nothing on its own.

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. From lesson 3: expand (x − 6)².
    Answerx² − 12x + 36: two strips of −6x and a corner of 36.
  2. Expand −3(x − 4).
    Answer−3x + 12: −3 × (−4) = +12.
  3. Simplify 5x² + 2x − 3x² + x.
    Answer2x² + 3x: collect the x² terms and the x terms separately.
Words used in this lesson
simplify
write with as few terms as possible. For example: 4x + 3 + x becomes 5x + 3.
substitute
put a number in place of a letter. For example: x = 2 in 3x + 1 gives 7.
equivalent
equal for every value of x. For example: (x + 1)² and x² + 2x + 1.
counter-example
one case that shows a claim is false. For example: x = 2 shows x² and 2x + 1 are different: 4 and 5.
expression
numbers and letters joined by operations, with no equals sign. For example: 3x² − x + 7.

Tomas or Priya?

Tomas and Priya both simplify (x + 5)(x − 2) − x(x + 1). Tomas gets 2x − 10. Priya gets 4x − 10.

Predict first: put x = 0 into the expression and into both answers. What do you notice? Then choose a better value to test, and decide who is right before you use the figure.

(x + 3)² − (x − 1)²
2
24the expression at this x
18the candidate at this x
0 of 9values from −4 to 4 where they agree

(x + 3)² − (x − 1)² and candidate B, 4x + 10, at x = 2: 24 and 18. They agree at 0 of the 9 values from −4 to 4.

Show me the steps
  1. (x + 3)² − (x − 1)² against candidate B, 4x + 10, at x = 2: 25 − 1 = 24 but 8 + 10 = 18. One value that disagrees proves a mistake.
  2. Candidate A, 8x + 8, at x = 2: 16 + 8 = 24. It agrees at all nine values from −4 to 4.
  3. 2(x + 4)(x − 3) against candidate A, 4x² + 4x − 48: they agree at x = 3 and at x = −4 and nowhere else from −4 to 4. Agreeing at one value is not enough.
  4. (x + 6)(x + 1) − (x + 2)(x + 3): candidate A, 2x, agrees at all nine values; candidate B, 12x + 12, agrees at none.

Expand, bracket, collect, check. Expand each product and keep each expansion in a bracket. A minus sign in front of a bracket changes the sign of every term inside it. Then collect like terms.

Check by putting the same value into the first line and the last. If they disagree, there is a mistake. If they agree, try another value: a slip can agree by luck. Avoid testing with only 0 (it hides slips in the x terms) or only 1 (it cannot tell x² from x).

For Tomas and Priya: x² + 3x − 10 − (x² + x) = 2x − 10, so Tomas is right. At x = 0 all three give −10, which is why 0 cannot settle it. At x = 3 the expression gives 8 − 12 = −4, Tomas gives −4 and Priya gives 2.

A common mistake: collecting unlike terms. 2x² + 3x cannot be written as 5x² or 5x³. At x = 2, 2x² + 3x = 14, but 5x² = 20 and 5x³ = 40.

Worked examples

Simplify (x + 4)(x + 2) − 3(x + 1).

  1. Expand: x² + 6x + 8 − 3x − 3.
  2. x² + 3x + 5.
  3. Check at x = 2: 6 × 4 − 3 × 3 = 15 and 4 + 6 + 5 = 15.

Simplify (x − 3)² − (x + 2)(x − 2).

  1. Expand and bracket: x² − 6x + 9 − (x² − 4).
  2. Remove the bracket: x² − 6x + 9 − x² + 4.
  3. −6x + 13. Check at x = 1: 4 − (−3) = 7 and −6 + 13 = 7.

Your go. The first step is given. Simplify (x + 5)(x + 1) − x(x + 2).

  1. Expand and bracket: x² + 6x + 5 − (x² + 2x).
  2. Now remove the bracket, collect, and check with a value that is not 0 or 1.
Check the answer

x² + 6x + 5 − x² − 2x = 4x + 5. Check at x = 3: 8 × 4 − 3 × 5 = 17 and 12 + 5 = 17.

Check before you go on

1. Which is (x + 4)(x − 1) − x² simplified?

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 + 3)(x + 5) − x² simplifies to px + q. What is p?

3. Find the value of (x + 2)(x − 7) when x = 3. Then check it by putting x = 3 into your expansion.

Level 2 · Building

4. (x + 6)² − (x + 2)(x + 10) simplifies to a single number. What is it?

5. Simplify 2(x − 5)(x + 5) − 2x². What number is left?

Level 3 · Stretch

6. For which value of x do (x + 5)(x − 3) and (x − 1)(x + 4) have the same value?

7. Expand and simplify (2x + 1)(x + 3) − (x + 1)². The result is ax² + bx + c. What is b?

Something to think about

Two expressions, each with x² as the highest power, agree at x = 1, at x = 2 and at x = 3. Must they be the same expression? Would your answer change if x³ were allowed?

One way to think about it

Take one from the other: what is left has x² at most, and is 0 at three values. An expression like that, if it is not 0 everywhere, can be 0 at two values at most (the figure's 2(x + 4)(x − 3) slip agrees at two). So the two must be the same. With x³ allowed, (x − 1)(x − 2)(x − 3) is 0 at 1, 2 and 3 but 6 at x = 4, so three agreements are no longer enough.

Exit ticket

8. Lin simplifies (x + 2)(x + 6) − x(x + 5) to 13x + 12. She tests x = 0 and both give 12. Which is true?