Year 9

Factorise quadratics of the form ${x}^{2}$ + bx + c

Factorise quadratic expressions of the form ${x}^{2}$ + bx + c into two brackets of the form (x + p)(x + q).

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Lesson overview

Algebra - expanding and factorising

Factorising a quadratic is the reverse of expanding double brackets. For ${x}^{2}$ + bx + c, you need two numbers p and q such that p + q = b (the coefficient of x) and p × q = c (the constant). Once you find p and q, write the factorised form as (x + p)(x + q). Always check by expanding back out.

Find two numbers that add to b and multiply to c

Factorise ${x}^{2}$ + 7x + 12.

Need: add to 7, multiply to 123 + 4 = 7 ✓

3 × 4 = 12 ✓

Factorised form(x + 3)(x + 4)

Check: expand to verify

List factor pairs of c (12): 1×12, 2×6, 3×4. Which pair adds to b (7)? → 3 and 4.

When c is positive and b is negative — both numbers are negative

Factorise ${x}^{2}$ − 8x + 15.

Need: add to −8, multiply to 15−3 + (−5) = −8 ✓

−3 × −5 = 15 ✓

Factorised form(x − 3)(x − 5)

When c is positive and b is negative, both numbers must be negative.

When c is negative — one number positive, one negative

Factorise ${x}^{2}$ + 2x − 15.

Need: add to 2, multiply to −155 + (−3) = 2 ✓

5 × (−3) = −15 ✓

Factorised form(x + 5)(x − 3)

When c is negative, one number is positive and one is negative. The larger (in size) determines the sign of b.

Using factorisation to solve quadratic equations

Solve ${x}^{2}$ + 5x + 6 = 0.

Eitherx + 2 = 0

→ x = −2

Orx + 3 = 0

→ x = −3

If two things multiply to zero, at least one of them must be zero. Set each bracket equal to zero and solve.

Worked example

Factorise ${x}^{2}$ − x − 12, then solve ${x}^{2}$ − x − 12 = 0.

Find: add to −1, multiply to −123 + (−4) = −1 and 3 × (−4) = −12

x + 3 = 0x = −3

x − 4 = 0x = 4

Find two numbers that multiply to −12 and add to −1.
Factor pairs of −12: try 3 and −4. Check: 3 + (−4) = −1 ✓ and 3 × (−4) = −12 ✓.
Write as (x + 3)(x − 4).
Set each bracket to zero: x + 3 = 0 gives x = −3; x − 4 = 0 gives x = 4.
Solutions: x = −3 or x = 4.

Try it

Use the examples carefully, then choose the answer.

Question 1

Factorise ${x}^{2}$ + 9x + 20.

💡 Find two numbers that add to 9 and multiply to 20.

Question 2

Factorise ${x}^{2}$ − 6x + 8.

💡 Both numbers must be negative (c is positive, b is negative).

Question 3

Factorise ${x}^{2}$ + 3x − 10.

💡 One number positive, one negative (c is negative). Which adds to 3?

Question 4

Solve ${x}^{2}$ + 6x + 5 = 0. Give both solutions.

💡 Factorise first: find two numbers adding to 6 and multiplying to 5.

Question 5

Factorise ${x}^{2}$ − 4. (Hint: this is a special case!)

💡 This is the difference of two squares. ${x}^{2}$ − 4 = ${x}^{2}$ + 0x − 4. Numbers: +2 and −2.

Factorise quadratics of the form x2{x}^{2}x2 + bx + c

Lesson overview

Worked example

Try it

Independent practice will plug in here

Factorise quadratics of the form ${x}^{2}$ + bx + c