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 + 3)(x + 5).
Answer
x² + 8x + 15. The x terms are 5x and 3x, and 3 × 5 = 15. - From Year 8: factorize 6x + 15.
Answer
3(2x + 5). 3 divides both terms. - Work out 7 × (−4) and 7 + (−4).
Answer
−28 and 3. The product of numbers of opposite sign is negative; the sum takes the smaller from the larger.
Words used in this lesson
- factorize
- write an expression as a product of brackets. For example: x² + 9x + 20 as (x + 4)(x + 5).
- expand
- multiply out brackets. For example: (x + 1)(x + 2) gives x² + 3x + 2.
- coefficient
- the number multiplying a letter. For example: the 7 in 7x.
- constant term
- the term with no letter. For example: the 12 in x² + 7x + 12.
- difference of two squares
- one square subtracted from another. For example: x² − 100.
- common factor
- a number or letter that divides every term. For example: 7 in 14x² + 21.
The tile panel
A craftsperson in Isfahan cuts a rectangular tile panel whose area is x² + 10x + 21 square units. Its two sides are both of the form x plus a whole number. Expanding two brackets gives a quadratic, so reading the quadratic backwards should give the brackets.
Predict first: two whole numbers multiply to 21 and add to 10. What are they, and so what are the two sides? Then use the figure to check the method on other numbers.
x² + 5x + 6: the pairs with product 6 are listed. 2 and 3 add to 5, so x² + 5x + 6 = (x + 2)(x + 3).
Show me the steps
- Slide c to 12 and b to 7. The pairs with product 12 include 3 and 4, which add to 7, so x² + 7x + 12 = (x + 3)(x + 4).
- Keep c = 12 and slide b to −8. The pair that adds to −8 is −2 and −6, so x² − 8x + 12 = (x − 2)(x − 6). Both numbers are negative because the product is positive and the sum is negative.
- Set c to −12 and b to 4. The product is negative, so one number is positive and one is negative. −2 and 6 add to 4, so x² + 4x − 12 = (x − 2)(x + 6).
- Set b = 3 and c = 4. The pairs with product 4 add to −5, −4, 4 and 5, never to 3. It does not factorize with whole numbers.
To factorize x² + bx + c, find two numbers with product c and sum b. They go in the brackets: (x + m)(x + n). If c is positive the numbers have the same sign; if c is negative they have opposite signs, and the one further from zero takes the sign of b.
Difference of two squares. a² − k² = (a + k)(a − k): the middle terms cancel, as the second figure shows.
When the x² has a number in front, first take out any common factor. Otherwise multiply the number in front by c, find the pair with that product and sum b, split the middle term into those two, then factorize in pairs.
a = 7, k = 2: the square has area 49, the corner has area 4, so the shape has area 49 − 4 = 45. Rearranged, it is a 9 by 5 rectangle: 9 × 5 = 45.
Show me the steps
- Set a = 5 and k = 3. The square has area 25 and the cut corner 9, so the shape has area 16. The rectangle is 8 wide and 2 tall: 8 × 2 = 16.
- Set a = 9 and k = 4. The area is 81 − 16 = 65, and the rectangle is 13 wide and 5 tall: 13 × 5 = 65.
- Set a = 6 and k = 1. The area is 36 − 1 = 35, and the rectangle is 7 wide and 5 tall: 7 × 5 = 35. Every setting gives the same area both ways.
Two common mistakes. (x + 4)² is not x² + 16: it is x² + 8x + 16, so the middle term is there. And x² + 9 is not (x + 3)²: (x + 3)² has a middle term 6x.
Worked examples
1. Factorize x² − 2x − 24.
- The product is −24 and the sum is −2, so the numbers have opposite signs and the negative one is further from zero.
- Try pairs of −24: 1 and −24 (sum −23), 2 and −12 (−10), 3 and −8 (−5), 4 and −6 (−2).
- x² − 2x − 24 = (x + 4)(x − 6). Check: x² − 6x + 4x − 24.
2. Factorize 25x² − 49.
- 25x² is (5x)² and 49 is 7², so this is a difference of two squares.
- 25x² − 49 = (5x + 7)(5x − 7).
3. Factorize 2x² + 7x + 3.
- Multiply the 2 by the 3 to get 6. Which pair has product 6 and sum 7? It is 1 and 6.
- Split the middle term: 2x² + x + 6x + 3.
- Factorize in pairs: x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
Your go. The first step is given. Factorize x² + 3x − 28.
- The product is −28, so one number is positive and one is negative, and the positive one is further from zero because the sum is +3.
- List the pairs of −28 and find the one that adds to 3.
Check the answer
7 and −4 have product −28 and sum 3, so the answer is (x + 7)(x − 4).
Check before you go on
1. Which is the factorized form of x² − 3x − 40?
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. x² + 11x + 18 = (x + a)(x + b), where a is less than b. Find b.
3. 25x² − 64 = (5x − m)(5x + m). Find m.
Level 2 · Building
4. 2x² + 11x + 5 = (2x + 1)(x + k). Find k.
5. A rectangular courtyard in Marrakech has area x² + 6x − 27 square metres. Its length is (x + 9) m and its width is (x − w) m. Find w.
Level 3 · Stretch
6. 6x² + 5x − 4 = (3x + p)(2x + q). Find p.
7. 3x² − 108 = 3(x − a)(x + a). Find a.
Something to think about
x² − 4 factorizes as (x − 2)(x + 2), but x² + 4 will not factorize with whole numbers. List the pairs with product 4 and look at their sums. What do you find, and why does no pair work?
One way to think about it
The pairs with product 4 are 1 and 4, 2 and 2, −1 and −4, −2 and −2, and their sums are 5, 4, −5 and −4. None is 0, and x² + 4 needs a pair that adds to 0. For x² − 4 the product is −4, and 2 and −2 add to 0. A difference of two squares has a middle term of 0 because the +ax and −ax cancel; a sum of squares has no such pair, and x² + 4 is always at least 4 so it never reaches 0 either.
Exit ticket
8. Which is the factorized form of 2x² − 5x − 3?