Quadratics: revision notes and practice
A quadratic has x² as its highest power. Simple factorising is GCSE Foundation extension content; factorising ax² + bx + c with a ≠ 1 is Higher content in AQA.
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What you will learn
- Recognise and expand quadratic expressions.
- Factorise x² + bx + c, including negative constants.
- Solve a factorised equation using the zero-product rule.
- Higher extensions: factorise non-monic quadratics and use the quadratic formula.
Key rules
Expand two brackets
Multiply each term in the first bracket by each term in the second. Collect the x terms: (x + p)(x + q) = x² + (p + q)x + pq.
Factorise x² + bx + c
Find two integers whose sum is b and whose product is c. Check by expanding your brackets.
Find both roots
Write the equation with zero on one side. If two factors multiply to zero, at least one factor is zero. Solve each factor separately.
Worked examples
Example 1: Factorise x² + x − 12.
- Find integers with sum 1 and product −12: 4 and −3.
- Use these in the brackets.
- Check: (x + 4)(x − 3) = x² − 3x + 4x − 12.
(x + 4)(x − 3)
Example 2: Higher extension: solve 2x² + 5x + 3 = 0.
- Factorise: (2x + 3)(x + 1) = 0.
- Check: the middle terms are 2x + 3x = 5x.
- Set 2x + 3 = 0 or x + 1 = 0, then solve each equation.
x = −1.5 or x = −1
A practice question
Expand (x − 5)(x + 6).
Check the answer and explanation
x² + 1x − 30
Multiply every term in the first bracket by every term in the second. Combine the x terms: 1x² + (6 + -5)x + (-5 × 6) = x² + 1x − 30.
1. Recognise a quadratic and the task
A quadratic expression has x² as its highest power and can be written ax² + bx + c with a ≠ 0. The coefficients a, b and c may be positive, negative or zero, except that a cannot be zero. The expression x² − 4x + 3 is quadratic, while 4x + 3 is linear.
The command changes what an answer should look like. Expand asks for multiplied-out terms, factorise asks for a product such as two brackets, and solve asks for values of x that satisfy an equation. A factorisation alone is not a complete solution to an equation.
2. Expand every pair of terms
When multiplying two brackets, every term in the first multiplies every term in the second. For (x + 2)(x + 5), the four products are x², 5x, 2x and 10. Collecting the middle terms gives x² + 7x + 10.
A grid with one bracket along each edge helps track all four products. With (x − 3)(x + 4), the middle terms are 4x and −3x, and the constant is −12, giving x² + x − 12. Preserve signs during multiplication before collecting.
(x + p)(x + q) = x² + (p + q)x + pq
3. Factorise when the x² coefficient is 1
For x² + bx + c, look for two numbers whose sum is b and whose product is c. To factorise x² + 7x + 12, choose 3 and 4, since they add to 7 and multiply to 12. The answer is (x + 3)(x + 4).
Start with factor pairs of the constant and test their sums. If no suitable integer pair exists, that does not prove the quadratic has no roots; it only means this integer-pair method has not produced factors. Check every proposed pair by expanding the brackets rather than matching only one coefficient.
4. Use signs and special forms
If c is negative, the two numbers have opposite signs. In x² − x − 12, −4 and 3 have product −12 and sum −1, giving (x − 4)(x + 3). If c is positive, the signs match: both positive for a positive sum, or both negative for a negative sum.
A difference of two squares has a special factorisation: x² − 25 = (x − 5)(x + 5). The middle terms cancel. A repeated factor also matters: x² + 6x + 9 = (x + 3)². If every term has a common factor, take it out first, as in 2x² − 8x = 2x(x − 4).
a² − b² = (a − b)(a + b)
5. Set factors equal to zero
The zero-product rule says that if a product equals zero, at least one factor is zero. From (x + 3)(x − 2) = 0, either x + 3 = 0 or x − 2 = 0, so x = −3 or x = 2. Include both distinct roots.
This rule applies to a product equal to zero, not to a product equal to another number. For example, (x + 3)(x − 2) = 5 does not justify setting a bracket equal to 5. Higher extension: if rearrangement is needed, first collect all terms on one side to form a quadratic equal to zero.
6. Higher extension: a non-monic quadratic
When a ≠ 1, guessing two constants alone is insufficient because the x terms also depend on the leading coefficients. One method splits the middle term. For 2x² + 7x + 3, find numbers with product ac = 6 and sum b = 7: they are 6 and 1.
Rewrite as 2x² + 6x + x + 3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Expanding verifies all three coefficients. If this equals zero, the roots are −1/2 and −3. Factorising ax² + bx + c with a ≠ 1 is Higher content in AQA.
7. Higher extension: the quadratic formula
The quadratic formula solves ax² + bx + c = 0 without needing simple integer factors. Identify a, b and c with their signs after putting the equation into that form. For x² − 2x − 1 = 0, a = 1, b = −2 and c = −1.
The expression under the square root is b² − 4ac = 8, so x = (2 ± √8)/2 = 1 ± √2. The plus and minus give two roots. If b² − 4ac is zero there is one repeated root; if it is negative there are no real roots. Use brackets carefully around the whole numerator and denominator when calculating.
x = (−b ± √(b² − 4ac))/(2a), for a ≠ 0
8. Check roots and interpret the context
Substitute each root into the original equation. For x² − x − 6 = 0, the roots 3 and −2 both make the left side zero. This catches a common sign error: a factor x − 3 gives root +3, while x + 2 gives root −2.
A graph of y = ax² + bx + c meets the x-axis at its real roots, because y is zero there. A repeated root corresponds to touching the axis rather than crossing it. In a length or time problem, a negative algebraic root may be unsuitable; show it first, then explain why the context rejects it. Keep exact roots unless rounding is requested.
Check your understanding
Expand (x − 2)(x + 5).
x² + 3x − 10.
Factorise x² − 7x + 10.
(x − 5)(x − 2), since −5 and −2 add to −7 and multiply to 10.
Solve x² + x − 6 = 0.
x = −3 or x = 2, from (x + 3)(x − 2) = 0.
Higher extension: solve 3x² − x − 2 = 0.
x = 1 or x = −2/3, from (3x + 2)(x − 1) = 0.
How to earn the marks
- Separate factorising an expression from solving an equation.
- Check both the product and sum in integer-pair factorising.
- Use the zero-product rule only after making one side zero.
- Show every distinct root and explain any contextual rejection.
A common mistake
A factorisation is an expression; roots solve an equation. Include both roots when they are different.
Course context and official specifications
This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.
- AQA GCSE Mathematics — A4, A18
Foundation practice uses monic quadratics and factorising to solve. Non-monic factorising, completing the square and the quadratic formula are Higher extensions; only the first is in this bank.
- Edexcel GCSE Mathematics — A4, A18
Foundation practice uses monic quadratics and factorising to solve. Non-monic factorising, completing the square and the quadratic formula are Higher extensions; only the first is in this bank.
- OCR GCSE Mathematics — 6.01, 6.03
Foundation practice uses monic quadratics and factorising to solve. Non-monic factorising, completing the square and the quadratic formula are Higher extensions; only the first is in this bank.
Keep your next step small
Try five questions, read the feedback, and revisit one thing you found difficult. The app tracks a 20-answer practice goal separately from accuracy.
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