IGCSE Mathematics: Quadratic Equations Practice Questions
A quadratic equation has the form ax2 + bx + c = 0 and usually has two solutions. Try factorising first, and use the quadratic formula when the expression does not factorise.
Topic 2.5 of Cambridge IGCSE Mathematics 0580 guarantees at least one quadratic on every Extended paper. The decision that matters is which method to use, and the question wording usually tells you. The four questions below cover all three methods.
What you need to know for Quadratic Equations
- Standard formRearrange to ax2 + bx + c = 0 before doing anything else. Every method assumes the equation equals zero.
- FactorisingFind two numbers multiplying to c and adding to b when a = 1. Set each bracket equal to zero and solve.
- The quadratic formulax = (minus b plus or minus the square root of (b2 minus 4ac)) divided by 2a. Use it whenever the expression will not factorise.
- Completing the squarex2 + bx + c becomes (x + b divided by 2)2 minus (b divided by 2)2 + c. Useful for finding turning points as well as solving.
- Choosing a methodGive exact answers or leave in surd form suggests completing the square or the formula. Give answers to 2 decimal places almost always means the formula.
- CheckingSubstitute both solutions back into the original equation. Both should give zero.
IGCSE Mathematics Quadratic Equations questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Solve x2 minus 7x + 12 = 0 by factorising.
Show the worked answer
- Find two numbers that multiply to give 12 and add to give minus 7.
- Both must be negative: minus 3 and minus 4 multiply to 12 and add to minus 7.
- Factorise: (x minus 3)(x minus 4) = 0.
- A product equals zero only if one factor is zero, so x minus 3 = 0 or x minus 4 = 0, giving x = 3 or x = 4.
Solve 2x2 + 5x minus 3 = 0 by factorising.
Show the worked answer
- Multiply the first and last coefficients: 2 times minus 3 = minus 6. Find two numbers multiplying to minus 6 and adding to 5, which are 6 and minus 1.
- Split the middle term: 2x2 + 6x minus x minus 3 = 0.
- Factorise in pairs: 2x(x + 3) minus 1(x + 3) = 0, so (2x minus 1)(x + 3) = 0.
- Set each bracket to zero: 2x minus 1 = 0 gives x = one half, and x + 3 = 0 gives x = minus 3.
Solve 3x2 minus 7x + 1 = 0, giving your answers correct to 3 significant figures.
Show the worked answer
- The expression does not factorise, so use the quadratic formula with a = 3, b = minus 7 and c = 1.
- Calculate the discriminant: b2 minus 4ac = 49 minus 12 = 37.
- x = (7 plus or minus the square root of 37) divided by 6. The square root of 37 is 6.0828.
- First solution: (7 + 6.0828) divided by 6 = 2.1805, which is 2.18. Second solution: (7 minus 6.0828) divided by 6 = 0.15280, which is 0.153.
Write x2 + 6x minus 2 in the form (x + p)2 + q, then use it to solve x2 + 6x minus 2 = 0, giving exact answers.
Show the worked answer
- Halve the coefficient of x: 6 divided by 2 = 3, so the bracket is (x + 3).
- (x + 3)2 expands to x2 + 6x + 9, which is 9 too much, so subtract 9: x2 + 6x = (x + 3)2 minus 9.
- Include the constant: x2 + 6x minus 2 = (x + 3)2 minus 9 minus 2 = (x + 3)2 minus 11.
- To solve, set it equal to zero: (x + 3)2 = 11, so x + 3 = plus or minus the square root of 11.
- x = minus 3 plus or minus the square root of 11. These are the exact answers, approximately 0.317 and minus 6.32.
Common mistakes in this topic
- Not rearranging to equal zero before solving.
- Giving only one root.
- Sign errors when substituting a negative b into the formula.
- Losing the plus or minus after taking a square root.
- Rounding partway through a formula calculation.
Exam tips
- Always rearrange to equal zero first. Every method depends on it.
- Try factorising for about thirty seconds. If nothing works, switch to the formula rather than persisting.
- Write a, b and c with their signs before substituting into the formula.
- If the question says exact or in surd form, do not use a decimal answer.
- Check both roots by substituting back. Both should give zero.
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Quadratic Equations FAQs
When should I use the quadratic formula instead of factorising?
Use factorising when two whole numbers can be found that multiply and add correctly. If the question asks for answers to a number of decimal places or significant figures, or if no factor pair works within about thirty seconds, switch to the formula.
What is the quadratic formula?
x equals minus b, plus or minus the square root of b squared minus 4ac, all divided by 2a. The equation must first be rearranged into the form ax squared plus bx plus c equals zero, and the signs of a, b and c must be carried through carefully.
How do I complete the square?
Halve the coefficient of x to get the number inside the bracket, square that bracket, then subtract the square of the halved value to compensate. Finally add the original constant term. The result gives both the turning point and an exact route to the solutions.
Why do quadratic equations have two solutions?
A quadratic graph is a parabola, which usually crosses the x-axis at two points, and each crossing is a solution. If the discriminant is zero there is one repeated root where the curve touches the axis, and if it is negative there are no real solutions.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.