Practice Maths

Factorising Trinomials

Key Ideas

Key Terms

monic trinomial
Has the form x2 + bx + c (leading coefficient is 1).
To factorise x2 + bx + c
Find two integers p and q such that
   p × q = c    and    p + q = b
   Then: x2 + bx + c = (x + p)(x + q)
Difference of two squares
A2 − b2 = (a + b)(a − b)
   Recognise this when you have two perfect square terms with a minus sign between them and no middle term.
Signs guide

   x2 + bx + c: both factors positive if c > 0 and b > 0
   x2 − bx + c: both factors negative if c > 0 and b > 0
   x2 + bx − c: factors have opposite signs; larger magnitude has same sign as b
Hot Tip Always check your factorisation by expanding your answer. If it gives back the original trinomial, you’re correct!

Worked Example

Question: Factorise x2 + 7x + 12.

Step 1 — Identify b and c.
b = 7 and c = 12

Step 2 — Find two numbers that multiply to 12 and add to 7.
Factor pairs of 12: (1, 12), (2, 6), (3, 4)
Check sums: 1+12=13, 2+6=8, 3+4=7 ✓

Step 3 — Write the factorised form.
x2 + 7x + 12 = (x + 3)(x + 4)

Check: (x+3)(x+4) = x2+4x+3x+12 = x2+7x+12 ✓

What Is a Monic Trinomial?

A trinomial is an expression with three terms, like x2 + 7x + 12. A monic trinomial has a leading coefficient of 1 (the x2 term has no number in front). Factorising monic trinomials is the main skill in this lesson.

The Factorising Method: Find Two Numbers

To factorise x2 + bx + c, you need two numbers that multiply to c and add to b. These two numbers become the constants in the two brackets.

x2 + 7x + 12: find two numbers that multiply to 12 and add to 7. Pairs that multiply to 12: (1, 12), (2, 6), (3, 4). The pair (3, 4) adds to 7. Answer: (x + 3)(x + 4).

x2 − 5x + 6: multiply to 6 and add to −5. Pairs: (−1, −6), (−2, −3). (−2) + (−3) = −5. Answer: (x − 2)(x − 3).

x2 + x − 12: multiply to −12 and add to 1. Pairs: (4, −3), (−4, 3), (6, −2), (−6, 2). 4 + (−3) = 1. Answer: (x + 4)(x − 3).

Choosing the Signs

Use the signs in the original trinomial to guide your choice:

If c is positive and b is positive: both numbers are positive.

If c is positive and b is negative: both numbers are negative.

If c is negative: one number is positive and one is negative (the larger one determines the sign of b).

Checking with FOIL and the Difference of Two Squares

Always verify by expanding with FOIL. If you get back to the original, your answer is correct.

The difference of two squares x2 − c2 is a special case: c = −c2 (negative), b = 0 (no middle term). This gives (x + c)(x − c). Example: x2 − 9 = (x + 3)(x − 3).

Exam strategy: When searching for the two numbers, write out factor pairs of c systematically rather than guessing. Start from the smallest pairs and work up. For x2 − 2x − 15, factors of −15: (1, −15), (−1, 15), (3, −5), (−3, 5). The pair (3, −5) adds to −2. Answer: (x + 3)(x − 5). Always check your factorisation by expanding!

Mastery Practice

  1. Factorise each monic trinomial. Fluency

    1. x2 + 5x + 6
    2. x2 + 8x + 15
    3. x2 − 6x + 8
    4. x2 − 9x + 14
    5. x2 + 3x − 10
    6. x2 − 2x − 15
    7. x2 + 11x + 30
    8. x2 − 7x + 12
    9. x2 + x − 12
    10. x2 − 4x − 21

  2. Factorise using the difference of two squares identity: a2 − b2 = (a + b)(a − b). Understanding

    1. x2 − 25
    2. y2 − 49
    3. 4a2 − 9
    4. 9m2 − 16
    5. 25p2 − 64
    6. x2 − 1

  3. Factorise these expressions. Some require taking out a common factor first. Understanding

    1. 2x2 + 10x + 12 (take out HCF first)
    2. 3x2 − 18x + 27 (take out HCF first)
    3. −x2 − 5x − 6 (take out −1 first)
    4. −x2 + 4x + 12 (take out −1 first)
    5. Verify your answer to part (a) by expanding (2x2 + 10x + 12 should be recovered).

  4. Solving quadratics and area problems by factorising. Problem Solving

    1. Solve x2 + 5x + 6 = 0 by:
      1. Factorising the left side.
      2. Setting each factor equal to zero.
      3. Writing the two solutions.
    2. Solve x2 − 4x − 12 = 0. Show all steps.
    3. The area of a rectangle is x2 + 9x + 20 cm2. By factorising, find expressions for the length and width. If x = 3, find the actual dimensions and area.
    4. A square has area (x2 − 36) cm2. Factorise to find the side length of the square. What value of x must be true for the area to be positive?
    5. Two positive integers differ by 4. Their product is 77. By writing the integers as x and x + 4 and forming a quadratic equation, find both integers.

  5. Factorise each non-monic trinomial (leading coefficient ≠ 1). Problem Solving

    Strategy. For ax2 + bx + c, find two numbers that multiply to a × c and add to b. Rewrite the middle term, then factor by grouping.
    1. 2x2 + 7x + 3
    2. 3x2 + 10x + 8
    3. 2x2 − 5x + 3
    4. 5x2 + 11x + 2
    5. 4x2 − 4x − 3
    6. 6x2 + x − 2

  6. Fully factorise each expression. Some require taking out a common factor first, others involve non-monic trinomials. Problem Solving

    1. 4x2 + 8x − 12
    2. 6x2 − 54
    3. 5x2 + 15x − 20
    4. 3x2 − 12x + 9
    5. 2x2 + 16x + 24
    6. 10x2 − 40

  7. Solve each quadratic equation by first fully factorising. Problem Solving

    1. 2x2 + 7x + 3 = 0
    2. 3x2 − 8x + 4 = 0
    3. 4x2 − 4x − 3 = 0
    4. 2x2 + 3x − 20 = 0
    5. 6x2 − 7x − 3 = 0

  8. Form and solve a quadratic equation for each problem. Problem Solving

    1. A rectangular garden has area (2x2 + 9x + 10) m2. Factorise to find expressions for the length and width. If x = 2, what are the actual dimensions and what is the area?
    2. A photo and its frame together have dimensions (3x + 1) cm by (x + 4) cm. Expanding confirms the total area is 3x2 + 13x + 4 cm2. Factorise to check, then find the dimensions when x = 5.
    3. The height h (in metres) of a ball thrown upward is modelled by h = −5t2 + 15t, where t is time in seconds. Factorise h to find when the ball is at ground level (h = 0). Interpret both solutions.

  9. Use the difference of two squares or other factorisation techniques to solve these problems. Problem Solving

    1. Compute 492 − 512 without a calculator by writing it as (49 + 51)(49 − 51).
    2. Show that n2 − 1 = (n + 1)(n − 1) for all integers n, and use this to explain why the product of any two consecutive even integers is always divisible by 8. (Hint: let n = 2k + 1.)
    3. Factorise 16x4 − 81y4 completely.
    4. Solve: x2 − 36 = 0. Explain why there are two solutions.

  10. Challenge: factorise and solve these extended problems. Problem Solving

    1. The product of two consecutive positive integers is 56. By writing the integers as x and x + 1, form and solve a quadratic equation to find both integers.
    2. A triangular sail has base (x + 5) m and height (x + 2) m. Its area is 24 m2. Form a quadratic equation and solve it to find x, then state the actual dimensions. (Recall: area of triangle = ½ base × height.)
    3. Determine the values of k for which x2 + kx + 9 factorises over the integers. (Hint: what must the two numbers p and q satisfy? List all integer factor pairs of 9 and find the possible sums.)
    4. Show algebraically that (x + a)2 − a2 = x2 + 2ax by expanding, and explain why this means x2 + 2ax always factorises as x(x + 2a).
See Answers ➔