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Factorization

Factorization

Factorising quadratics

Three Terms Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. When the x² has no number in front
  4. When the x² has a number in front
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.2: factorising quadratics
  • 2.2Factorise \(x^{2}+bx+c\) with any signs.
  • 2.2Factorise \(ax^{2}+bx+c\) by splitting the middle term.
  • 2.2Take out a common factor first (including x in a cubic), and spot perfect squares.

Before you start: Common factors and the difference of two squares. Next: Factorising by grouping.

Formula bank

Simple quadraticLearn it\(x^{2}+bx+c=(x+p)(x+q)\)\(p\times q=c\) and \(p+q=b\).Use it when the \(x^{2}\) has no number in front.
Splitting the middle termLearn it\(ax^{2}+bx+c\): find \(m, n\) with \(mn=ac\), \(m+n=b\)then \(ax^{2}+mx+nx+c\) and group in pairs.Use it when a number in front of \(x^{2}\).
Perfect squaresLearn it\(a^{2}+2ab+b^{2}=(a+b)^{2}\), \(a^{2}-2ab+b^{2}=(a-b)^{2}\)first and last terms are squares; the middle is 2 × their product.

When the x² has no number in front

\((x+p)(x+q)=x^{2}+(p+q)x+pq\). So to factorise \(x^{2}+bx+c\) find two numbers that multiply to c and add to b. The signs tell you where to look: c positive → both numbers have b's sign; c negative → opposite signs.

General methodHow to factorise \(x^{2}+bx+c\)

  1. List factor pairs of c (ignore signs for now).
  2. Pick the pair that gives b: c positive → ADD the pair; c negative → SUBTRACT them.
  3. Give the numbers signs so they multiply to c and add to b.
  4. Write \((x+p)(x+q)\) and expand to check the middle term.

Example Factorise \(x^{2}-3x-28\).

Multiply to −28, add to −3: 7 and 4 differ by 3, the bigger one is negative.

  1. numbers \(-7\) and \(+4\)
  2. check: \(-7x+4x=-3x\) ✓

Answer \((x-7)(x+4)\)

Try it yourself

1
Factorise \(x^{2}+5x+6\).
Show answer

Answer \((x+2)(x+3)\)

2
Factorise \(x^{2}-2x-15\).
Show answer

Answer \((x-5)(x+3)\)

3
Factorise \(x^{2}-10x+21\).
Hint
c positive, b negative: both negative.
Show answer

Answer \((x-3)(x-7)\)

When the x² has a number in front

General methodHow to factorise \(ax^{2}+bx+c\) (splitting the middle term)

  1. Take out any common factor first.
  2. Work out \(a\times c\).
  3. Find two numbers that multiply to \(ac\) and add to \(b\).
  4. Split the middle term into those two: \(ax^{2}+mx+nx+c\).
  5. Factorise the first pair and the last pair; the brackets will match.
  6. Take out the matching bracket. Expand to check.

Example Factorise \(3x^{2}+10x-8\).

\(ac=-24\); numbers that add to 10: 12 and −2.

  1. \(3x^{2}+12x-2x-8\)
  2. \(3x(x+4)-2(x+4)\)
  3. \((3x-2)(x+4)\)

Answer \((3x-2)(x+4)\)

Look out for: a common factor (\(2x^{2}+10x+12=2(x+2)(x+3)\)), a cubic with \(x\) in every term (\(2x^{3}-x^{2}-15x=x(2x+5)(x-3)\)), a perfect square (\(9x^{2}-12x+4=(3x-2)^{2}\)) and a negative \(x^{2}\) (\(12+x-x^{2}=(4-x)(3+x)\)).

Try it yourself

1
Factorise \(2x^{2}+7x+3\).
Hint
ac = 6: numbers 6 and 1.
Show answer

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

2
Factorise \(2x^{2}-5x-3\).
Hint
ac = −6: numbers −6 and 1.
Show answer

Answer \((2x+1)(x-3)\)

3
Factorise \(x^{2}-14x+49\).
Hint
49 = 7², and 14 = 2 × 7.
Show answer

Answer \((x-7)^{2}\)

Ways they test it

Type 1 Factorise a quadratic

"Factorise \(x^{2}-3x-28\)." (2 marks)

Spot it three terms, no common factor.

  1. × to c, + to b (or ac when a ≠ 1).
  2. Expand to check.

Type 2 Factorise completely

"Factorise completely \(2x^{2}-8x-42\)." (2 marks)

Spot it 'completely': a common factor is hiding.

  1. Take out the HCF.
  2. Factorise the quadratic left.

For example \(2x^{2}-8x-42\)

  1. \(2(x^{2}-4x-21)\)

Answer \(2(x-7)(x+3)\)

Type 3 A cubic

"Factorise \(3x^{3}+3x^{2}-18x\)." (3 marks, 4024 Paper 1)

Spot it every term has x.

  1. Take out \(3x\).
  2. Factorise the quadratic.

For example \(3x^{3}+3x^{2}-18x\)

  1. \(3x(x^{2}+x-6)\)

Answer \(3x(x+3)(x-2)\)

Type 4 Two letters

"Factorise \(6x^{2}-13xy+6y^{2}\)."

Spot it \(xy\) in the middle, \(y^{2}\) at the end.

  1. Treat \(y\) like a number.
  2. Split the middle as usual.

For example \(6x^{2}-13xy+6y^{2}\)

  1. \(6x^{2}-9xy-4xy+6y^{2}\)
  2. \(3x(2x-3y)-2y(2x-3y)\)

Answer \((3x-2y)(2x-3y)\)

Type 5 Factorise, then use it

"Factorise ... Hence solve ..." or "Simplify the fraction ..."

Spot it 'hence' after a factorise part.

  1. Use your brackets straight away (next pages).

Mistakes to avoid

  • ✗ \(x^{2}-x-12=(x+4)(x-3)\)✓ \((x-4)(x+3)\)That gives +x: check the middle term's sign.
  • ✗ \(2x^{2}+7x+3=(2x+3)(x+1)\)✓ \((2x+1)(x+3)\)Right end numbers, wrong places: expand to check.
  • ✗ \(2x^{2}+10x+12=(2x+4)(x+3)\)✓ \(2(x+2)(x+3)\)Not fully: 2 is common in a bracket.

Summary

  • \(x^{2}+bx+c\): two numbers that × to c and + to b.
  • \(ax^{2}+bx+c\): × to ac, + to b, split the middle, group in pairs.
  • Take out a common factor first (x too, in a cubic).
  • Always expand your answer to check the middle term.

If you see …

  • c is negativethe two numbers have opposite signs.
  • a number in front of x²use a × c and split the middle term.

Practice

Practice: easy, medium, hard

Easy

1
Factorise \(x^{2}+7x+12\).
Show answer

Answer \((x+3)(x+4)\)

2
Factorise \(x^{2}+8x+15\).
Show answer

Answer \((x+3)(x+5)\)

3
Factorise \(x^{2}+2x-24\).
Hint
6 and 4 differ by 2.
Show answer

Answer \((x+6)(x-4)\)

Medium

1
Factorise \(6x^{2}-x-2\).
Guide me

What is it asking? two brackets

What do we know? \(ac=-12\)

Plan numbers −4 and 3, then split

Show working
  1. \(6x^{2}-4x+3x-2\)
  2. \(2x(3x-2)+1(3x-2)\)
  3. \((3x-2)(2x+1)\)

Answer \((3x-2)(2x+1)\)

2
Factorise completely \(2x^{2}-8x-42\).
Guide me

What is it asking? a product

What do we know? 2 is common

Plan then \(x^{2}-4x-21\)

Show working
  1. \(2(x^{2}-4x-21)\)
  2. \(2(x-7)(x+3)\)

Answer \(2(x-7)(x+3)\)

3
Factorise \(9x^{2}-12x+4\).
Guide me

What is it asking? a product

What do we know? \(9x^{2}=(3x)^{2}\), \(4=2^{2}\)

Plan middle \(=2\times3x\times2\): perfect square

Show working
  1. \((3x-2)(3x-2)=(3x-2)^{2}\)

Answer \((3x-2)^{2}\)

Hard

1
Factorise \(5x^{2}+13x-6\).
Guide me

What is it asking? two brackets

What do we know? \(ac=-30\)

Plan 15 and −2

Show working
  1. \(5x^{2}+15x-2x-6\)
  2. \(5x(x+3)-2(x+3)\)
  3. \((5x-2)(x+3)\)

Answer \((5x-2)(x+3)\)

PartAnswer / workingMarksGuidance
\(5x^{2}+15x-2x-6\) or \((5x+a)(x+b)\) with \(ab=-6\)M1
\((5x-2)(x+3)\)A1
Total: 2 marks
2
Factorise completely \(3x^{3}+3x^{2}-18x\).
Guide me

What is it asking? a product

What do we know? \(3x\) is in every term

Plan then \(x^{2}+x-6\)

Show working
  1. \(3x(x^{2}+x-6)\)
  2. \(3x(x+3)(x-2)\)

Answer \(3x(x+3)(x-2)\)

PartAnswer / workingMarksGuidance
\(3x(x^{2}+x-6)\)M1
\(3x(x+3)(x-2)\)A1
Total: 2 marks
3
Factorise \(4x^{2}+4x-15\).
Guide me

What is it asking? two brackets

What do we know? \(ac=-60\)

Plan 10 and −6

Show working
  1. \(4x^{2}+10x-6x-15\)
  2. \(2x(2x+5)-3(2x+5)\)
  3. \((2x+5)(2x-3)\)

Answer \((2x+5)(2x-3)\)

PartAnswer / workingMarksGuidance
\(4x^{2}+10x-6x-15\)M1
\((2x+5)(2x-3)\)A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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