The quadratic formula solves any equation of the form ax² + bx + c = 0 by substituting the coefficients directly into x = (−b ± √(b² − 4ac)) / 2a. It is most useful when a quadratic cannot be factorised neatly, and it always works — whether the solutions are integers, fractions, or surds.
What is the quadratic formula?
For any quadratic equation written in the standard form ax² + bx + c = 0, the two solutions are:
x = (−b ± √(b² − 4ac)) / 2a
The ± symbol means you calculate two values: one using addition (+ √) and one using subtraction (− √). These two values are the roots of the quadratic — the x-values where the parabola crosses the x-axis.
The formula is given on the AQA, Edexcel and OCR GCSE formula sheets, so you do not need to memorise it. What you must be able to do is read it accurately, substitute correctly, and carry out the arithmetic without error.
When should you use the formula instead of factorising?
Three situations call for the quadratic formula:
| Situation | Example | Method |
|---|---|---|
| Factorises easily with integers | x² + 5x + 6 = 0 | Factorise: (x+2)(x+3) = 0 |
| Does not factorise (irrational roots) | x² + 3x + 1 = 0 | Quadratic formula |
| Question asks for surd or decimal answers | 3x² − 2x − 7 = 0 | Formula always works |
| Question says "give your answer to 2 d.p." | Any quadratic | Formula, then round at the end |
A reliable test: try to find two integers whose product equals ac and whose sum equals b. If no such pair exists within a few seconds, switch to the formula.
How do you use the quadratic formula step by step?
Worked example: Solve 2x² + 5x − 12 = 0.
- Identify a, b, and c: a = 2, b = 5, c = −12.
- Calculate the discriminant: b² − 4ac = 5² − 4 × 2 × (−12) = 25 + 96 = 121.
- Take the square root: √121 = 11.
- Calculate the two solutions:
- x = (−5 + 11) / (2 × 2) = 6/4 = 3/2
- x = (−5 − 11) / (2 × 2) = −16/4 = −4
- Verify: 2(3/2)² + 5(3/2) − 12 = 2(9/4) + 15/2 − 12 = 9/2 + 15/2 − 12 = 24/2 − 12 = 12 − 12 = 0. ✓
Second example (surd answers): Solve x² − 4x + 1 = 0. Give your answers in surd form.
- a = 1, b = −4, c = 1.
- Discriminant: (−4)² − 4(1)(1) = 16 − 4 = 12.
- √12 = √(4 × 3) = 2√3.
- x = (−(−4) ± 2√3) / (2 × 1) = (4 ± 2√3) / 2 = 2 ± √3.
What is the discriminant and what does it tell you?
The expression b² − 4ac inside the square root is called the discriminant. Its value reveals how many real solutions the quadratic has before you do any further calculation:
| Discriminant | Number of real roots | Graph behaviour |
|---|---|---|
| b² − 4ac > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| b² − 4ac = 0 | One repeated root (equal roots) | Parabola just touches the x-axis |
| b² − 4ac < 0 | No real roots | Parabola does not cross the x-axis |
If you calculate the discriminant and it is negative, stop — the equation has no real solutions. Do not try to take the square root of a negative number at GCSE.
How do you avoid calculator errors with the formula?
- Calculate −b first. If b = −5, then −b = +5. Writing −(−5) clearly prevents sign errors.
- Square b before negating. (−4)² = 16, not −16. The squaring always gives a positive result.
- Evaluate the discriminant as a single number before rooting it. Write b² − 4ac = … on its own line.
- Divide the entire numerator by 2a. A common error is dividing only the ±√(discriminant) part, leaving −b undivided.
- Use brackets on your calculator. Enter (−b + √(discriminant)) ÷ (2a) with explicit brackets to make the calculator follow the correct order.
Frequently asked questions
Does the quadratic formula always give exact answers?
It gives exact answers in the sense that the algebra is exact. However, if the discriminant is not a perfect square, the answers involve surds (e.g. 2 ± √3). The question usually signals whether to leave answers in surd form or to round to a given number of decimal places.
What if the equation is not in standard form?
Rearrange it first. For example, 3x² = 7x − 1 must be rewritten as 3x² − 7x + 1 = 0 before you can read off a = 3, b = −7, c = 1. Always ensure the right-hand side is zero before identifying the coefficients.
Is the quadratic formula given on the exam paper?
Yes — the AQA, Edexcel and OCR GCSE exam formula sheets all include x = (−b ± √(b² − 4ac)) / 2a. You do not need to memorise it. However, you do need to read it carefully from the sheet rather than relying on a half-remembered version, because errors most commonly come from misremembering a minus sign or the position of 2a.
How do you check your answers once you have them?
Substitute each answer back into the original equation. If x = 3/2 is a root of 2x² + 5x − 12 = 0, then 2(3/2)² + 5(3/2) − 12 must equal exactly zero. If it does not, either the arithmetic or the coefficient identification was wrong. Checking both roots in full earns you the method marks even if one answer was wrong.
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