Quadratic equations with three solving methods
Choose factorisation, completing the square or the quadratic formula, then check every real solution.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Factorise quadratic expressions.
- Complete the square and simplify square roots.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If (x − 4)(x + 2) = 0, what are the roots?
What you will learn
- Use the zero-product rule after factorisation.
- Solve by completing the square and by the quadratic formula.
- Use the discriminant and context to interpret roots.
Put the equation in standard form
A quadratic equation can be written ax² + bx + c = 0 with a ≠ 0. Bring every term to one side before identifying a, b and c. For x(x + 2) = 15, the standard form is x² + 2x − 15 = 0. The constant is −15, including its sign.
If a product is zero, at least one factor is zero. Factorising gives (x + 5)(x − 3) = 0, so x = −5 or 3. The rule requires a zero right side. For (x + 5)(x − 3) = 8, setting either factor to zero would not solve the stated equation.
A completed square gives exact roots
For x² − 4x − 1 = 0, write (x − 2)² = 5. Then x = 2 ± √5. Both signs matter because both a positive and negative value of x − 2 can have square five. Keep the surd form when an exact answer is requested.
If the completed square equals zero, the root is repeated. If it equals a negative number, there are no real roots. A context may restrict otherwise valid roots: a length must be positive, while an unrestricted mathematical equation may allow a negative solution. Discard a root only with an explicit reason from the model.
The quadratic formula works beyond easy factor pairs
For ax² + bx + c = 0, use x = [−b ± √(b² − 4ac)]/(2a). The denominator divides the complete numerator. Substitute negative coefficients in brackets. For 2x² + 3x − 1 = 0, the roots are (−3 ± √17)/4.
The discriminant D = b² − 4ac tells how many real roots occur: two distinct when D > 0, one repeated when D = 0 and none when D < 0. It also checks whether an apparent factorisation is plausible. Factorisation is efficient when factors are evident; completed-square form helps interpret a turning point; the formula is systematic for any quadratic with real coefficients.
Pause and explain
How many real roots does x² + 2x + 5 = 0 have?
Worked example
Solve 2x² − 5x − 3 = 0 by factorisation and verify the roots.
Show the worked solution
- Factorise as (2x + 1)(x − 3) = 0.
- The zero-product rule gives x = −1/2 or x = 3.
- At −1/2 the expression is 1/2 + 5/2 − 3 = 0; at three it is 18 − 15 − 3 = 0.
Answer x = −1/2 or 3
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve x² − 7x + 10 = 0.
Give me a hint
Find a pair with product ten and sum negative seven.
Compare my reasoning
- The pair is −5 and −2.
- Write (x − 5)(x − 2) = 0.
- The roots are 5 and 2; substitute each to check.
x = 2 or 5
Look for these in your work
- Matched the factor pair to both coefficients.
- Used the zero-product rule.
Solve x² − 4x − 1 = 0 exactly.
Give me a hint
Complete the square, or use a = 1, b = −4 and c = −1.
Compare my reasoning
- Completing gives (x − 2)² = 5.
- Take both roots: x − 2 = ±√5.
- Thus x = 2 ± √5; the formula gives the same result.
x = 2 ± √5
Look for these in your work
- Retained exact surd answers.
- Included both possible signs.
A rectangle has width x metres and length x + 3 metres. Its area is 28 m². Find its dimensions.
Give me a hint
Build an area equation before choosing which root is meaningful.
Compare my reasoning
- x(x + 3) = 28 gives x² + 3x − 28 = 0.
- Factorise as (x + 7)(x − 4) = 0, so x = −7 or 4.
- A width must be positive, giving width 4 m and length 7 m; 4 × 7 = 28.
4 m by 7 m
Look for these in your work
- Constructed the quadratic from area.
- Rejected the negative root using the physical context.
Common mistakes
- Using the zero-product rule when the product is not zero.
- Losing the negative sign of b or c in the formula.
- Discarding a negative mathematical root without a contextual restriction.
What can you explain now?
Solve 3x² + x − 2 = 0.
Compare with the explanation
x = −1 or 2/3
(3x − 2)(x + 1) = 0, so each factor gives one root.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Solve one quadratic by two methods and compare the exact answers. Explain what its discriminant says before calculating any decimals.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.