Sheet № 20 · Higher only · AQA · Edexcel · OCR
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
§Key definitions
Question:
Solve x² + 6x + 2 = 0, giving your answers in surd form.
Answer:
x = −3 + √7 or x = −3 − √7
Check:
2(2.77)² − 3(2.77) − 7 = 15.3458 − 8.31 − 7 = 0.0358 ≈ 0 ✓
Q1:
Solve x² + 4x − 3 = 0, giving your answers in simplified surd form.
Q2:
Solve 3x² + 7x + 1 = 0, giving your answers to 2 decimal places.
§Formulas to memorise
x = (−b ± √(b² − 4ac)) / 2a
Discriminant = b² − 4ac
b² − 4ac > 0 — → two distinct real solutions.
b² − 4ac = 0 — → one repeated real solution.
b² − 4ac < 0 — → no real solutions.
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 −.
Worked example
Solve x² + 6x + 2 = 0, giving your answers in surd form.
Working:
⚠ 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.