Chapter 2 of 8

All 8 chapters
  1. 1Expanding two brackets, readunderstood
  2. 2Factorising quadratics, readunderstood
  3. 3Using factorising, readunderstood
  4. 4Completing the square, readunderstood
  5. 5The quadratic formula, readunderstood
  6. 6Graphs of quadratics, readunderstood
  7. 7Quadratic sequences, readunderstood
  8. 8A line and a curve, readunderstood

M4.9.2 Chapter 2

Factorising quadratics

I can expand (x + 3)(x + 4) to get x² + 7x + 12. How do I go backwards?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Factorise x² + 7x + 12
  • Factorise a difference of two squares, like x² − 49
  • Factorise 2x² + 7x + 3

To factorise is to write an expression as a product of brackets. It undoes expanding. The brackets are the factors of the expression.

Look again at the rectangle for x² + 7x + 12. It is x + 4 long and x + 3 wide. The parts hold x², 4x, 3x and 12.

A rectangle cut into four parts. The top is marked x + 4 and is cut into x and 4. The left side is cut into x and 3. The four parts hold x², 4x, 3x and 12.x4x3x + 4x²4x3x12
The two numbers 4 and 3 multiply to the last part, 12. They add to the x parts, 4x + 3x = 7x.

This is the key. In x² + 7x + 12, the last number 12 is the product of the two numbers in the brackets. The x number 7 is their sum. So look for two numbers with product 12 and sum 7.

A table with the heads pair, product and sum. Rows: 1, 12 with product 12 and sum 13; 2, 6 with product 12 and sum 8; 3, 4 with product 12 and sum 7. The last row is marked.pairproductsum1, 1212132, 61283, 4127
List the pairs with product 12. Only the pair 3 and 4 has sum 7.

Draw it first

From x² + 7x + 12 to two brackets

  1. A column of expressions with arrows between them. The first is x² + 7x + 12. Step 2 adds an arrow with the note find two numbers, product 12 and sum 7, and the line 3 and 4. Step 3 adds an arrow with the note put them in brackets, and (x + 3)(x + 4).x² + 7x + 12

    1 Start with the quadratic, x² + 7x + 12.

  2. A column of expressions with arrows between them. The first is x² + 7x + 12. Step 2 adds an arrow with the note find two numbers, product 12 and sum 7, and the line 3 and 4. Step 3 adds an arrow with the note put them in brackets, and (x + 3)(x + 4).x² + 7x + 12x² + 7x + 123 and 4product 12, sum 7

    2 Find two numbers with product 12 and sum 7. They are 3 and 4.

  3. A column of expressions with arrows between them. The first is x² + 7x + 12. Step 2 adds an arrow with the note find two numbers, product 12 and sum 7, and the line 3 and 4. Step 3 adds an arrow with the note put them in brackets, and (x + 3)(x + 4).x² + 7x + 12x² + 7x + 123 and 4product 12, sum 7x² + 7x + 123 and 4(x + 3)(x + 4)product 12, sum 7put them in brackets

    3 Write each number in a bracket with x: (x + 3)(x + 4).

Worked example

Factorise x² + 7x + 12.

  1. The last number is 12 and the x number is 7. Find two numbers with product 12 and sum 7.
  2. The pairs with product 12 are 1 and 12, 2 and 6, and 3 and 4. Their sums are 13, 8 and 7.
  3. Only the pair 3 and 4 adds up to 7: 3 + 4 = 7.
  4. So x² + 7x + 12 = (x + 3)(x + 4).
  5. Check by expanding: x² + 4x + 3x + 12 = x² + 7x + 12.

Answer: (x + 3)(x + 4)

The signs tell you what kind of numbers to look for. Read the last number first, and then the x number.

A table with the heads last, x number and brackets. Rows: +, +, both +; +, −, both −; −, + or −, one of each.lastx numberbrackets++both ++−both −−+ or −one of each
When the last number is negative, the two numbers have different signs. The one further from 0 takes the sign of the x number.

Worked example

Factorise x² − 5x + 6.

  1. The last number is +6, so the two numbers have the same sign. The x number is −5, so both are negative.
  2. Try −2 and −3. Their product is (−2) × (−3) = 6.
  3. Their sum is (−2) + (−3) = −5, which is the x number.
  4. So x² − 5x + 6 = (x − 2)(x − 3).
  5. Check by expanding: x² − 3x − 2x + 6 = x² − 5x + 6.

Answer: (x − 2)(x − 3)

Worked example

Factorise x² + 2x − 15.

  1. The last number is −15, so one number is positive and one is negative. The x number is +2, so the positive one is further from 0.
  2. The pairs with product 15 are 1 and 15, and 3 and 5. Make the one further from 0 positive: 15 and −1, or 5 and −3.
  3. Try 5 and −3. Their product is 5 × (−3) = −15.
  4. Their sum is 5 + (−3) = 2, which is the x number.
  5. So x² + 2x − 15 = (x − 3)(x + 5).
  6. Check with x = 1: (1 − 3)(1 + 5) = −12, and 1 + 2 − 15 = −12.

Answer: (x − 3)(x + 5)

Some quadratics have no x term, like x² − 49. The number 49 is a square, since 7 × 7 = 49. Try the brackets (x + 7)(x − 7) in a grid.

A grid for (x + 7)(x − 7). The terms x and +7 are across the top. The terms x and −7 are down the side. The cells hold x², +7x, −7x and −49. The cells +7x and −7x are hatched, and under the grid is = x² − 49.x+7x−7x²+7x−7x−49= x² − 49
The hatched cells are +7x and −7x. They cancel, so only x² and −49 are left.

This is a difference of two squares: one square minus another. It always factorises the same way. a² − b² = (a + b)(a − b). The middle terms cancel because the signs are opposite.

Worked example

Factorise x² − 49.

  1. The number 49 is a square, so x² − 49 is a difference of two squares. The square root of 49 is 7.
  2. Use two brackets with +7 in one and −7 in the other: (x + 7)(x − 7).
  3. Check by expanding: x² − 7x + 7x − 49 = x² − 49. The middle terms cancel.

Answer: (x + 7)(x − 7)

It works only when there is a minus between two squares. The expression x² + 49 has no brackets of this kind, because no real numbers multiply to 49 and add to 0.

When the number in front of x² is not 1, a grid helps. The quadratic 2x² + 7x + 3 comes from the brackets (2x + 1)(x + 3). Here is the grid.

A grid for (2x + 1)(x + 3). The terms 2x and +1 are across the top. The terms x and +3 are down the side. The cells hold 2x², +x, +6x and +3. The cells +6x and +x are hatched, and under the grid is = 2x² + 7x + 3.2x+1x+32x²+x+6x+3= 2x² + 7x + 3
The two hatched cells, 6x and x, add to the middle term 7x. They multiply to 6x², which is 2x² × 3.

Worked example

Factorise 2x² + 7x + 3.

  1. Multiply the first number and the last number: 2 × 3 = 6.
  2. Find two numbers with product 6 and sum 7. Try 1 and 6: their product is 6 and their sum is 1 + 6 = 7.
  3. Split the middle term: 2x² + x + 6x + 3.
  4. Take out a common factor from each pair: x(2x + 1) + 3(2x + 1).
  5. Both parts have the factor (2x + 1), so the answer is (2x + 1)(x + 3).
  6. Check with x = 1: (2 + 1)(1 + 3) = 12, and 2 + 7 + 3 = 12.

Answer: (2x + 1)(x + 3)

Always look for a common factor first. In 3x² + 12x + 9 every term divides by 3. So 3x² + 12x + 9 = 3(x² + 4x + 3) = 3(x + 1)(x + 3).

See it three ways

With things

Lay out tiles: one big square x², eight strips of x, and 15 small squares. Arrange them into one rectangle.

As a picture

A rectangle cut into four parts. The top is marked x + 5 and is cut into x and 5. The left side is cut into x and 3. The four parts hold x², 5x, 3x and 15.x5x3x + 5x²5x3x15

In words

The last number 15 is the product of the two numbers 3 and 5. The x number 8 is their sum. So the brackets are x + 3 and x + 5.

In symbols

x² + 8x + 15 = (x + 3)(x + 5).

Try it

A difference of two squares at work

You need: paper and a pencil.

  1. Work out 51 × 49 on paper. It takes a while.
  2. Now think of x = 50. Then 51 × 49 is (x + 1)(x − 1), which is x² − 1.
  3. Work out 50 × 50 − 1. Compare the two answers.

What you will see: Both ways give 2 499. A difference of two squares turns a hard product into a square minus a small number.

If you cannot try it, this is what happens.
A table with the heads sum and answer. Rows: 51 × 49 gives 2 499; 50 × 50 − 1 gives 2 499.sumanswer51 × 492 49950 × 50 − 12 499

A wrong idea many people have

x² − 49 = (x − 7)(x − 7).

Why it feels right: The number 7 times 7 is 49, so it seems that two matching brackets should work.

The test that breaks it: Put x = 10. Then x² − 49 = 51, but (10 − 7)(10 − 7) = 9. The two numbers differ.

What is true: The two brackets need opposite signs, so that the middle terms cancel. The answer is (x + 7)(x − 7).

A wrong idea many people have

x² − 5x + 6 = (x + 2)(x + 3), because 2 × 3 = 6 and 2 + 3 = 5.

Why it feels right: The numbers 2 and 3 do multiply to 6 and add to 5, so they look right.

The test that breaks it: Expand it: (x + 2)(x + 3) = x² + 5x + 6. The middle term is +5x, but the quadratic has −5x.

What is true: The x number is −5, so the two numbers must add to −5. Use −2 and −3. Always expand to check the signs.

Key words

factorise
to write an expression as a product of brackets.
factor
a bracket or number that divides an expression exactly.
difference of two squares
an expression like x² − 49: one square minus another square.
common factor
a factor that every term has, such as 3 in 3x² + 12x + 9.

Back to your guess

The answer is C: Find two numbers that multiply to 12 and add to 7.

In (x + 3)(x + 4) the last numbers multiply: 3 × 4 = 12. The x terms add: 3x + 4x = 7x. So going backwards, find two numbers that multiply to 12 and add to 7. They are 3 and 4.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Factorise x² + 9x + 18.

    Tap your answer.

    1. Not this one. The numbers 1 and 18 multiply to 18, but they add to 19, not 9. Go to M4.9.2. Try another.

    2. Yes, that is right. Find two numbers with product 18 and sum 9. The pair 3 and 6 works: 3 × 6 = 18 and 3 + 6 = 9. Check: x² + 6x + 3x + 18 = x² + 9x + 18.

    3. Not this one. These multiply to 18, but they add to −9. The x number is +9, so both numbers are positive. Go to M4.9.2. Try another.

    4. Not this one. These add to 9, but they multiply to 20, not 18. Go to M4.9.2. Try another.

    See the answer

    (x + 3)(x + 6)

    Find two numbers with product 18 and sum 9. The pair 3 and 6 works: 3 × 6 = 18 and 3 + 6 = 9. Check: x² + 6x + 3x + 18 = x² + 9x + 18.

    • If you chose A: The numbers 1 and 18 multiply to 18, but they add to 19, not 9. Go to M4.9.2.
    • If you chose C: These multiply to 18, but they add to −9. The x number is +9, so both numbers are positive. Go to M4.9.2.
    • If you chose D: These add to 9, but they multiply to 20, not 18. Go to M4.9.2.

  2. Factorise x² − 7x + 10.

    Tap your answer.

    1. Not this one. These multiply to 10 and add to +7. The x number is −7, so both numbers are negative. Go to M4.9.2. Try another.

    2. Not this one. These multiply to 10, but they add to −11, not −7. Go to M4.9.2. Try another.

    3. Yes, that is right. The last number is +10 and the x number is −7, so both numbers are negative. The pair −2 and −5 works: (−2) × (−5) = 10 and (−2) + (−5) = −7.

    4. Not this one. These multiply to −10, not +10. The last number is +10, so both numbers have the same sign. Go to M4.9.2. Try another.

    See the answer

    (x − 2)(x − 5)

    The last number is +10 and the x number is −7, so both numbers are negative. The pair −2 and −5 works: (−2) × (−5) = 10 and (−2) + (−5) = −7.

    • If you chose A: These multiply to 10 and add to +7. The x number is −7, so both numbers are negative. Go to M4.9.2.
    • If you chose B: These multiply to 10, but they add to −11, not −7. Go to M4.9.2.
    • If you chose D: These multiply to −10, not +10. The last number is +10, so both numbers have the same sign. Go to M4.9.2.

  3. Factorise x² + 3x − 10.

    Tap your answer.

    1. Not this one. These multiply to +10. The last number is −10, so one number is positive and one is negative. Go to M4.9.2. Try another.

    2. Not this one. These multiply to −10, but they add to −3. The x number is +3, so the number further from 0 must be the positive one. Go to M4.9.2. Try another.

    3. Not this one. These multiply to −10, but they add to −9, not 3. Go to M4.9.2. Try another.

    4. Yes, that is right. The last number is −10, so the numbers have different signs. The x number is +3, so the one further from 0 is positive. The pair −2 and 5 works: (−2) × 5 = −10 and (−2) + 5 = 3.

    See the answer

    (x − 2)(x + 5)

    The last number is −10, so the numbers have different signs. The x number is +3, so the one further from 0 is positive. The pair −2 and 5 works: (−2) × 5 = −10 and (−2) + 5 = 3.

    • If you chose A: These multiply to +10. The last number is −10, so one number is positive and one is negative. Go to M4.9.2.
    • If you chose B: These multiply to −10, but they add to −3. The x number is +3, so the number further from 0 must be the positive one. Go to M4.9.2.
    • If you chose C: These multiply to −10, but they add to −9, not 3. Go to M4.9.2.

  4. Factorise x² − 81.

    Tap your answer.

    1. Yes, that is right. The number 81 is 9 × 9, so x² − 81 is a difference of two squares. Two brackets with +9 and −9 make the middle terms cancel: x² − 9x + 9x − 81.

    2. Not this one. Expand it: x² − 18x + 81. The middle terms must cancel, so one bracket needs +9 and the other −9. Go to M4.9.2. Try another.

    3. Not this one. Expand it: x² + 18x + 81. The middle terms must cancel, so the brackets need opposite signs. Go to M4.9.2. Try another.

    4. Not this one. It can. The number 81 is 9 × 9, so x² − 81 is a difference of two squares. Go to M4.9.2. Try another.

    See the answer

    (x + 9)(x − 9)

    The number 81 is 9 × 9, so x² − 81 is a difference of two squares. Two brackets with +9 and −9 make the middle terms cancel: x² − 9x + 9x − 81.

    • If you chose B: Expand it: x² − 18x + 81. The middle terms must cancel, so one bracket needs +9 and the other −9. Go to M4.9.2.
    • If you chose C: Expand it: x² + 18x + 81. The middle terms must cancel, so the brackets need opposite signs. Go to M4.9.2.
    • If you chose D: It can. The number 81 is 9 × 9, so x² − 81 is a difference of two squares. Go to M4.9.2.

  5. Factorise 2x² + 5x + 2.

    Tap your answer.

    1. Not this one. Expand it: 2x² + 2x + 2x + 2 = 2x² + 4x + 2. The middle term is 4x, not 5x. Go to M4.9.2. Try another.

    2. Not this one. Expand it: x² + 3x + 2. The 2 in front of x² is missing. Go to M4.9.2. Try another.

    3. Yes, that is right. Multiply 2 × 2 = 4. The numbers 1 and 4 have product 4 and sum 5. Split the middle: 2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (2x + 1)(x + 2).

    4. Not this one. Expand it: 2x² − 5x + 2. The middle term is −5x, but the quadratic has +5x. Go to M4.9.2. Try another.

    See the answer

    (2x + 1)(x + 2)

    Multiply 2 × 2 = 4. The numbers 1 and 4 have product 4 and sum 5. Split the middle: 2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (2x + 1)(x + 2).

    • If you chose A: Expand it: 2x² + 2x + 2x + 2 = 2x² + 4x + 2. The middle term is 4x, not 5x. Go to M4.9.2.
    • If you chose B: Expand it: x² + 3x + 2. The 2 in front of x² is missing. Go to M4.9.2.
    • If you chose D: Expand it: 2x² − 5x + 2. The middle term is −5x, but the quadratic has +5x. Go to M4.9.2.

  6. Priya factorises a quadratic. How can she check her brackets?

    Tap your answer.

    1. Not this one. Two brackets do not make an answer right. Wrong brackets can also come in twos. Expand them to see. Go to M4.9.2. Try another.

    2. Yes, that is right. Factorising undoes expanding, so expanding must undo factorising. If the expansion does not match, there is a mistake.

    3. Not this one. The two numbers multiply to the last number and add to the x number. Expanding the brackets is the full check. Go to M4.9.2. Try another.

    See the answer

    Expand the brackets. She should get the quadratic she started with.

    Factorising undoes expanding, so expanding must undo factorising. If the expansion does not match, there is a mistake.

    • If you chose A: Two brackets do not make an answer right. Wrong brackets can also come in twos. Expand them to see. Go to M4.9.2.
    • If you chose C: The two numbers multiply to the last number and add to the x number. Expanding the brackets is the full check. Go to M4.9.2.

  7. Factorise fully 5x² + 15x + 10.

    Tap your answer.

    1. Not this one. It expands to 5x² + 15x + 10, so it is right, but not fully factorised. The bracket 5x + 5 still has a common factor, 5. Go to M4.9.2. Try another.

    2. Not this one. Expand it: x² + 3x + 2. The common factor 5 outside the brackets is missing. Go to M4.9.2. Try another.

    3. Not this one. The common factor 5 divides every term. Dividing 10 by 5 gives 2, not 10. Go to M4.9.2. Try another.

    4. Yes, that is right. Every term divides by 5: 5x² + 15x + 10 = 5(x² + 3x + 2). Then x² + 3x + 2 = (x + 1)(x + 2), since 1 × 2 = 2 and 1 + 2 = 3.

    See the answer

    5(x + 1)(x + 2)

    Every term divides by 5: 5x² + 15x + 10 = 5(x² + 3x + 2). Then x² + 3x + 2 = (x + 1)(x + 2), since 1 × 2 = 2 and 1 + 2 = 3.

    • If you chose A: It expands to 5x² + 15x + 10, so it is right, but not fully factorised. The bracket 5x + 5 still has a common factor, 5. Go to M4.9.2.
    • If you chose B: Expand it: x² + 3x + 2. The common factor 5 outside the brackets is missing. Go to M4.9.2.
    • If you chose C: The common factor 5 divides every term. Dividing 10 by 5 gives 2, not 10. Go to M4.9.2.

  8. From an earlier chapter

    Expand (x + 7)(x − 2).

    Tap your answer.

    1. Not this one. The x terms are 7x and −2x. Add the numbers: 7 + (−2) = 5, so the term is +5x. Go to M4.9.1. Try another.

    2. Yes, that is right. The four products are x², −2x, 7x and −14. The x terms join: −2x + 7x = 5x. So the answer is x² + 5x − 14.

    3. Not this one. The last product is 7 × (−2). A positive times a negative is negative, so it is −14. Go to M4.9.1. Try another.

    4. Not this one. The x terms are 7x and −2x. Add the numbers: 7 + (−2) = 5, not 9. Go to M4.9.1. Try another.

    See the answer

    x² + 5x − 14

    The four products are x², −2x, 7x and −14. The x terms join: −2x + 7x = 5x. So the answer is x² + 5x − 14.

    • If you chose A: The x terms are 7x and −2x. Add the numbers: 7 + (−2) = 5, so the term is +5x. Go to M4.9.1.
    • If you chose C: The last product is 7 × (−2). A positive times a negative is negative, so it is −14. Go to M4.9.1.
    • If you chose D: The x terms are 7x and −2x. Add the numbers: 7 + (−2) = 5, not 9. Go to M4.9.1.

  9. Find two numbers that multiply to 20 and add to 9. Give the smaller one first.

    See the answer

    4 and 5

    The pairs with product 20 are 1 and 20, 2 and 10, and 4 and 5. Only 4 + 5 = 9.

Keep it

Copy this into your exercise book.

Find two numbers that multiply to the last number and add to the x number. Check the signs. Look for a common factor first. Expand to check.

Draw: The table of pairs for 12 with the pair 3 and 4 marked, and the brackets (x + 3)(x + 4).

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (898 KB). It opens on any phone.