The quadratic formula, explained with examples

By The STEM-Tutor team · Updated October 2026

The quadratic formula solves any quadratic equation, even ones that will not factorise. It is a Higher tier GCSE skill and is used all the way through A level.

The formula

For any quadratic equation in the form ax² + bx + c = 0, with a ≠ 0, the solutions are

x = (−b ± √(b² − 4ac)) / 2a

The ± sign means there are usually two answers: one using + and one using −. For GCSE Maths exams in 2025, 2026 and 2027, a formulae sheet is provided with the papers, covering formulae students previously had to memorise; check the sheet for your exam year, but learn the formula anyway, because recalling it quickly saves time. A common way to remember it is to say it aloud as a rhythm: “minus b, plus or minus the square root of b squared minus four a c, all over two a”.

When to use it

There are three main ways to solve a quadratic: factorising, completing the square and the quadratic formula. Factorising is quickest when it works, but many quadratics do not factorise with whole numbers. The formula always works. Use it when:

• the question asks for answers to a number of decimal places or significant figures, which is a strong hint the quadratic does not factorise;
• you have tried to factorise for a short while and nothing fits;
• the coefficients are awkward, such as decimals or large numbers.

A step-by-step method

1. Rearrange the equation so that it equals zero: ax² + bx + c = 0.
2. Write down a, b and c, including their signs.
3. Work out the discriminant b² − 4ac first, on its own line.
4. Substitute into the formula, using brackets around negative numbers.
5. Work out the two answers separately, then round as the question asks.

Calculating the discriminant separately is the single best habit for avoiding errors, because most mistakes happen inside the square root.

Worked examples

Example 1: a straightforward case

Solve x² + 5x + 3 = 0, giving answers to 2 decimal places.

• a = 1, b = 5, c = 3.
• Discriminant: 5² − 4 × 1 × 3 = 25 − 12 = 13.
• x = (−5 ± √13)/2.
• x ≈ −0.70 or x ≈ −4.30.

Example 2: a negative coefficient

Solve 2x² − 3x − 4 = 0, giving answers to 3 significant figures.

• a = 2, b = −3, c = −4.
• Discriminant: (−3)² − 4 × 2 × (−4) = 9 + 32 = 41.
• x = (3 ± √41)/4, because −b = −(−3) = 3.
• x ≈ 2.35 or x ≈ −0.851.

Example 3: rearranging first

Solve 3x² = 7x − 1, to 2 decimal places.

• Rearrange: 3x² − 7x + 1 = 0, so a = 3, b = −7, c = 1.
• Discriminant: 49 − 12 = 37.
• x = (7 ± √37)/6, giving x ≈ 2.18 or x ≈ 0.15.

Example 4: answers in surd form

Solve x² − 4x + 1 = 0, leaving your answer in surd form.

• Discriminant: 16 − 4 = 12, and √12 = 2√3.
• x = (4 ± 2√3)/2 = 2 ± √3. Simplify the surd and cancel the common factor.

Example 5: a word problem

A rectangle has length (x + 3) cm and width x cm. Its area is 20 cm². Find x to 2 decimal places.

• x(x + 3) = 20, so x² + 3x − 20 = 0.
• Discriminant: 9 + 80 = 89. x = (−3 ± √89)/2, giving 3.22 or −6.22.
• A length cannot be negative, so x ≈ 3.22 cm. Always check answers make sense in context.

Example 6: an equation with fractions

Solve 1/x + 1/(x + 2) = 1, to 2 decimal places.

• Multiply through by x(x + 2): (x + 2) + x = x(x + 2), so 2x + 2 = x² + 2x.
• That gives x² = 2, so x ≈ ±1.41. Not every problem needs the full formula once it is simplified.

The discriminant: how many solutions?

The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions there are before you finish the calculation:

• b² − 4ac > 0: two different real solutions. The graph crosses the x-axis twice.
• b² − 4ac = 0: one repeated solution. The graph just touches the x-axis.
• b² − 4ac < 0: no real solutions, because you cannot take the square root of a negative number. The graph does not reach the x-axis.

If the discriminant is a perfect square, such as 25 or 49, the quadratic would have factorised. That is a useful check. The discriminant is used much more at A level, for example to find when a line is a tangent to a curve.

Where the formula comes from

The quadratic formula is what you get by completing the square on the general equation ax² + bx + c = 0. Divide by a, move c/a across, add (b/2a)² to both sides, and take square roots. You do not need to reproduce the proof at GCSE, but knowing it comes from completing the square helps you see that the two methods always agree.

Common mistakes

• Using the wrong sign for b, especially when b is negative: −b of −3 is +3.
• Writing −3² on a calculator instead of (−3)², which gives −9 instead of 9.
• Dividing only the square root by 2a, instead of the whole numerator.
• Forgetting to rearrange to = 0 before reading off a, b and c.
• Rounding the discriminant or the square root too early.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is the quadratic formula given in the GCSE exam?

For GCSE exams in 2025, 2026 and 2027, a formulae sheet is provided with the Maths papers, covering formulae students previously had to memorise. Check the sheet for your exam year. Learn the formula anyway: recalling it instantly saves time and avoids copying errors.

Is the quadratic formula on the Foundation paper?

No. It is a Higher tier topic. Foundation students solve quadratics by factorising.

Can I use my calculator's equation solver?

Some calculators solve quadratics directly, which is useful for checking. But you must show the formula and substitution to earn method marks, and it will not help on the non-calculator paper.

Key takeaways

• x = (−b ± √(b² − 4ac))/2a.
• Rearrange to = 0 first and read a, b and c with their signs.
• Work out the discriminant on its own line.
• The discriminant tells you how many solutions there are.
• Check answers make sense in context, rejecting negative lengths.

Related guides

• Completing the square
• Factorising quadratics
• GCSE Maths surds