Quadratic Equations: Factorising, the Formula and Completing the Square
A quadratic equation is one where the highest power of the unknown is x^2, written as ax^2 + bx + c = 0, and is solved by factorising, using the quadratic formula, or completing the square. Completing the square rewrites it as (x + p)^2 + q, which also reveals the turning point of the graph. All three methods are required across AQA Further Maths papers, often within larger problems.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Check whether the quadratic can be factorised by finding two numbers that multiply to give ac and add to give b, then split the middle term and factorise in pairs.
- If it does not factorise easily, use the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a), substituting the values of a, b and c carefully.
- To complete the square, halve the coefficient of x to form (x + b/2)^2, then subtract (b/2)^2 and add the original constant term.
- Set each factor (or the completed square expression) equal to zero and solve for x to find the root(s).
- Use the discriminant b^2 - 4ac to check the number of real roots before solving in full: positive gives two roots, zero gives one repeated root, negative gives none.
- Check your solution(s) by substituting back into the original equation.
Worked example
Solve the equation x^2 - 2x - 24 = 0 by factorising.
- Find two numbers that multiply to give -24 and add to give -2: these are -6 and 4.
- Write the factorised form: (x - 6)(x + 4) = 0.
- Set each factor equal to zero: x - 6 = 0 or x + 4 = 0.
- Solve each equation: x = 6 or x = -4.
- Final answer: x = 6 or x = -4.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Factorise x^2 + 9x + 20.Show answer
Answer: (x + 4)(x + 5)
Q2Solve x^2 + 4x - 21 = 0 by factorising.Show answer
Answer: x = -7 or x = 3
Q3Factorise fully 6x^2 + x - 2.Show answer
Answer: (2x - 1)(3x + 2)
Q4Factorise fully 3x^2 - 27.Show answer
Answer: 3(x - 3)(x + 3)
Q5By completing the square, write x^2 + 8x + 5 in the form (x + p)^2 + q.Show answer
Answer: (x + 4)^2 - 11 (p = 4, q = -11)
Q6Solve 2x^2 + 3x - 6 = 0 using the quadratic formula, giving your answers correct to 3 significant figures.Show answer
Answer: x = 1.14 or x = -2.64 (3 s.f.)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Solve the equation x^2 - x - 20 = 0 by factorising. Show your working.
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By completing the square, express 2x^2 - 8x + 3 in the form a(x + p)^2 + q, where a, p and q are constants.
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Solve the equation x^4 - 10x^2 + 9 = 0.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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