Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: expand (x + 4)(x − 9).
Answer
x² − 5x − 36. The x terms are 4x and −9x, and 4 × (−9) = −36. - From Year 9: expand (2x − 3)(x + 6).
Answer
2x² + 9x − 18. The x terms are 12x and −3x, and (−3) × 6 = −18. - From Year 10, Lesson 1: what value of x makes the denominator of 1 ÷ (x + 8) equal to 0?
Answer
−8. x + 8 = 0 gives x = −8.
Words used in this lesson
- common factor
- a number or letter that divides every term. For example: 6 in 6x² + 18x.
- difference of two squares
- one square minus another, which factorizes. For example: x² − 100 = (x + 10)(x − 10).
- perfect square
- a quadratic that is a bracket squared. For example: x² + 18x + 81 = (x + 9)².
- coefficient
- the number multiplying a letter. For example: the 8 in 8x.
- split the middle term
- write bx as two terms with product ac and sum b. For example: 9x as 6x + 3x.
- factorize completely
- keep going until no bracket has a common factor. For example: 2x(x + 3), not x(2x + 6).
The tile that must fit its frame
A potter in Oaxaca cuts a rectangular tile whose area, in square centimetres, is 6x² + 7x + 2 for a length x. The two sides are expressions that multiply to give the area. She wants the two side lengths as expressions in x.
Predict first: what could the two sides be, and which numbers in 6x² + 7x + 2 would you use to find them? Then use the figure to see how the brackets and the zeros come from a, b and c.
x² + 5x + 6: ac = 6, the pair 2 and 3 has product 6 and sum 5, so it factorizes as (x + 2)(x + 3). Zeros: −3 and −2.
Show me the steps
- Keep a = 1 and set b = 7 and c = 12. The pair with product 12 and sum 7 is 3 and 4, so the brackets are (x + 3)(x + 4) and the zeros are −4 and −3.
- Set a = 1, b = 1 and c = −12. The pair now has product −12 and sum 1: 4 and −3. The brackets are (x + 4)(x − 3), and the zeros are −4 and 3.
- Set a = 3, b = 10 and c = 8. Now ac = 24 and the pair is 4 and 6, so 10x splits into 4x + 6x and 3x² + 4x + 6x + 8 groups as (3x + 4)(x + 2). The zeros are −2 and −4/3.
- Set a = 3, b = 5 and c = 4. Now ac = 12 and no pair multiplies to 12 and adds to 5, because 1 + 12, 2 + 6 and 3 + 4 give 13, 8 and 7. It does not factorize with whole numbers, and the graph never reaches the x-axis.
Common factor first. 4x² − 36 = 4(x² − 9), and then the brackets. Taking it out first makes the numbers smaller.
The method for ax² + bx + c. Find two whole numbers m and n with m × n = a × c and m + n = b. Split the middle term, ax² + mx + nx + c, and group in pairs. If no such pair exists, there are no whole-number brackets.
Why it works. (px + q)(rx + s) = pr x² + (ps + qr)x + qs. The two middle pieces are ps and qr, and their product is ps × qr = pr × qs = a × c, while their sum is b. So the pair must have product ac and sum b, and any brackets give one.
Patterns. A difference of two squares, A² − B² = (A + B)(A − B). A perfect square, A² + 2AB + B² = (A + B)². Both are special cases of the same method.
A sum of two squares does not factorize. x² − 100 = (x + 10)(x − 10), but x² + 100 has no whole-number brackets. And (x + 9)² is x² + 18x + 81, not x² + 81: the middle term is 2 × 9 × x.
Worked examples
1. Factorize 6x² − 7x − 20.
- ac = 6 × (−20) = −120, and b = −7. The pair with product −120 and sum −7 is −15 and 8.
- Split: 6x² − 15x + 8x − 20 = 3x(2x − 5) + 4(2x − 5).
- Factor out the common bracket: (2x − 5)(3x + 4). Check: 6x² + 8x − 15x − 20.
2. Factorize 18x² − 50.
- Common factor first: 18x² − 50 = 2(9x² − 25).
- 9x² = (3x)² and 25 = 5², so a difference of two squares: 9x² − 25 = (3x + 5)(3x − 5).
- The full factorization is 2(3x + 5)(3x − 5).
3. Does 3x² + 8x + 6 factorize with whole numbers?
- ac = 18, b = 8. The pairs with product 18 are 1 and 18, 2 and 9, 3 and 6, with sums 19, 11 and 9.
- None adds to 8, and the negative pairs add to negative numbers.
- It does not factorize with whole numbers. (Its graph never reaches the x-axis.)
Your go. The first step is given. Factorize 4x² + 12x + 9.
- ac = 4 × 9 = 36 and b = 12, so the pair is 6 and 6.
- Now split 12x as 6x + 6x and group in two pairs.
Check the answer
4x² + 6x + 6x + 9 = 2x(2x + 3) + 3(2x + 3) = (2x + 3)(2x + 3) = (2x + 3)². It is a perfect square, which is why the pair is a repeated number.
Check before you go on
1. Which is the complete factorization of 7x² − 28?
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. 2x² + 29x + 14 = (2x + 1)(x + p). Find p.
3. 25x² − 49 = (5x + k)(5x − k). Find k.
Level 2 · Building
4. 6x² + 23x − 13 = (2x − 1)(3x + m). Find m.
5. A rectangular strip of kente cloth has area (12x² + 11x − 5) cm² and length (4x + 5) cm. Find its width, in centimetres, when x = 2.
Level 3 · Stretch
6. x² + kx + 36 is a perfect square for two values of k. What is the positive value?
7. For how many whole numbers k with 1 ≤ k ≤ 12 does x² + kx + 18 factorize into brackets with whole numbers?
Something to think about
x² − 25 factorizes as (x + 5)(x − 5), but x² + 25 will not factorize with whole numbers. Use the pair idea (product 25, sum 0) to say why. Then decide whether 4x² + 25 can be factorized.
One way to think about it
For x² + 25 you need two numbers with product 25 and sum 0. Numbers with sum 0 are opposites, m and −m, and their product is −m², which is never positive. So no pair exists. The same reasoning gives 4x² + 25: the pair would need product 100 and sum 0, which is again −m², so it does not factorize with whole numbers. A sum of two squares has no real brackets; a difference does.
Exit ticket
8. Which of these does NOT factorize into brackets with whole numbers?