Solving Quadratic Equations Using the Quadratic Formula

When a quadratic equation does not factorise neatly, the quadratic formula is your go-to method. It works for every quadratic equation, producing exact solutions that may be left in surd form or rounded to a given number of decimal places. This topic is exclusive to the Higher tier and appears regularly on AQA, Edexcel, and OCR papers. On this page you will learn the formula itself, how to identify a, b, and c, how to substitute carefully, and how to interpret the discriminant. If factorising is a scalpel, the quadratic formula is a Swiss army knife — it always works.

What Is the Quadratic Formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

The symbol ± means "plus or minus," which is why you get two solutions (one using +, one using −).

The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions exist:

b² − 4ac > 0 → two distinct real solutions.
b² − 4ac = 0 → one repeated real solution.
b² − 4ac < 0 → no real solutions.

Discriminant = b² − 4ac

The quadratic formula is given on the AQA and Edexcel formula sheets, but you should aim to memorise it so you do not waste time looking it up under exam pressure.

Step-by-Step Method

Rearrange the equation into the form ax² + bx + c = 0.
Identify a, b, and c. Be very careful with negative signs. For example, in 3x² − 5x + 1 = 0, a = 3, b = −5, c = 1.
Substitute into the formula. Write out the full substitution before simplifying — this reduces errors and earns method marks.
Calculate the discriminant (b² − 4ac) separately to keep your working tidy.
Simplify the numerator. You will have two versions: one with + and one with −.
Divide by 2a to find each solution.
Give your answer in the form requested: exact (surd) form, or rounded to a specified number of decimal places.

Worked Example 1 — Standard Application

Question: Solve x² + 6x + 2 = 0, giving your answers in surd form.

Working:

Here a = 1, b = 6, c = 2.

Step 1: Discriminant = b² − 4ac = 36 − 8 = 28.

Step 2: Substitute into the formula.

x = (−6 ± √28) / 2(1) = (−6 ± √28) / 2

Step 3: Simplify √28 = √(4 × 7) = 2√7.

x = (−6 ± 2√7) / 2 = −3 ± √7

Answer: x = −3 + √7 or x = −3 − √7

Worked Example 2 — Higher Level with Rounding

Question: Solve 2x² − 3x − 7 = 0, giving your answers correct to 2 decimal places.

Working:

Here a = 2, b = −3, c = −7.

Step 1: Discriminant = (−3)² − 4(2)(−7) = 9 + 56 = 65.

Step 2: Substitute.

x = (−(−3) ± √65) / 2(2) = (3 ± √65) / 4

Step 3: √65 ≈ 8.0623.

x₁ = (3 + 8.0623) / 4 = 11.0623 / 4 ≈ 2.77

x₂ = (3 − 8.0623) / 4 = −5.0623 / 4 ≈ −1.27

Check: 2(2.77)² − 3(2.77) − 7 = 15.3458 − 8.31 − 7 = 0.0358 ≈ 0 ✓

Answer: x ≈ 2.77 or x ≈ −1.27

Common Mistakes

Getting the sign of b wrong. In −b, if b is already negative, then −b is positive. Write out "−(−3) = 3" explicitly.
Forgetting that 2a means 2 × a, not 2 + a. If a = 3, then 2a = 6, and you divide the entire numerator by 6.
Not dividing the entire numerator by 2a. Students sometimes divide only part of the expression. Use a fraction line under the whole of −b ± √(b² − 4ac).
Rounding too early. Keep the full value of √(discriminant) until the final step. Rounding intermediate values introduces error.
Failing to simplify surds when asked for exact form. √28 should be written as 2√7. Practise surd simplification separately if needed.

Exam Tips

Write out "a = ..., b = ..., c = ..." before substituting. This earns you a mark on most papers and helps prevent sign errors.
Calculate the discriminant first, on its own line. This makes your working clearer and catches the case where b² − 4ac < 0 (no real solutions).
If the question says "give your answers to 2 d.p.," use the formula — this is a hint that the quadratic does not factorise neatly.
Memorise the formula even though it is on the formula sheet. Under time pressure, being able to write it from memory saves valuable seconds. See our GCSE Maths Formulas You Must Know for a complete list.

Practice Questions

Q1: Solve x² + 4x − 3 = 0, giving your answers in simplified surd form.

Answer: x = −2 ± √7

Q2: Solve 3x² + 7x + 1 = 0, giving your answers to 2 decimal places.

Answer: x ≈ −0.15 or x ≈ −2.18

Q3: Use the discriminant to determine how many real solutions 2x² − 4x + 5 = 0 has.

Answer: Discriminant = 16 − 40 = −24. Since −24 < 0, the equation has no real solutions.

Need to practise the quadratic formula until it becomes second nature? Start revising with GCSEMathsAI — our AI tutor provides step-by-step guidance and checks every line of your working.

For an overview of all quadratic methods, read our blog post on How to Solve Quadratic Equations at GCSE.

Summary

The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any quadratic equation.
Always rearrange to ax² + bx + c = 0 first and identify a, b, c carefully.
The discriminant b² − 4ac tells you how many real solutions exist.
Show full substitution to earn method marks.
Simplify surds when the question asks for exact answers.
Do not round intermediate values — only round at the final step.
The quadratic formula complements factorising and completing the square; choose the method that best fits the question.

Quadratic Formula –