Solve any quadratic equation with x=2a−b±b2−4ac, giving roots to 2 decimal places or as simplified surds. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Every quadratic ax2+bx+c=0 can be solved with the formula x=2a−b±b2−4ac. Factorising is faster when it works, but the formula always works (over the real numbers, as long as the discriminant is not negative). The exam's hint that you need it is usually 'give your answers to 2 decimal places' or 'in the form rp±q'.
The expression under the square root, b2−4ac, is the discriminant. Positive: two real roots. Zero: one repeated root. Negative: no real roots — say so, do not invent a square root of a negative. Write a, b and c with their signs, put brackets around every negative, and remember that the whole of −b±… is divided by 2a.
If the quadratic is not already =0, rearrange first. Completing the square is the same mathematics in a different layout — use it when you also need the turning point, and use the formula when you just need the two x-values quickly.
Objectives
Substitute a, b and c into x = (-b ± √(b² - 4ac)) / (2a).
Simplify the surd and cancel a common factor.
State both roots, or say when the discriminant is negative.
Key points
Rearrange into ax2+bx+c=0 and list a, b, c with signs.
Evaluate the discriminant b2−4ac on its own line.
Substitute into the formula with brackets around negatives.
Give both roots, rounded as asked or left in simplified surd form.
Worked examples
Example 1
Use the quadratic formula to give the exact roots of x² - 2x - 1 = 0.
2 marks
a = 1, b = -2, c = -1
Discriminant 4 + 4 = 8
(2 ± √8) / 2 = 1 ± √2
Answer: 1 ± √2
Common slips
Slip: b is 2 because of the minus The sign of b stays with the coefficient.
Slip: Discriminant 4 - 4 = 0 Minus 4ac, and c is already negative, so you add 4.
Slip: (2 ± √8) / 2 = 2 ± √2 Divide 2 and 2√2 by 2. √8 = 2√2.
Example 2
The quadratic formula is needed for 2x² + 3x - 1 = 0. Give the exact roots.
2 marks
a = 2, so the denominator is 4
Discriminant 9 + 8 = 17
(-3 ± √17) / 4
Answer: (-3 ± √17)/4
Common slips
Slip: Divide by 2 because a is 2 The formula divides by 2a, which is 4.
Slip: Discriminant 9 - 8 = 1 4ac is 4 times 2 times -1, so subtracting it adds 8.
Slip: -3 ± √17 The denominator 4 has to stay.
Example 3
Give the exact roots of x² - 3 = 0. There is no x term, so b is 0.
3 marks
a = 1, b = 0, c = -3
Discriminant 0 + 12 = 12
±√12 / 2 = ±√3
Answer: ±√3
Common slips
Slip: c is 3 The constant is negative.
Slip: Discriminant 3 4ac is -12, and subtracting a negative adds.
Slip: ±√12 √12 = 2√3, then divide by 2.
Questions and answers
GCSE Maths the quadratic formula questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Give the exact roots of x² + 4x - 1 = 0 by the quadratic formula.
Show answer
Answer: -2 ± √5
b is 4 and c is -1, so the discriminant is 20.
(-4 ± 2√5) / 2 simplifies to -2 ± √5.
M1 — b is 4 and c is -1, so the discriminant is 20.
A1 — (-4 ± 2√5) / 2 simplifies to -2 ± √5.
Question 2 Easy2 marks
State the exact roots of x² - 4x - 1 = 0. Use the quadratic formula rather than decimals.
Show answer
Answer: 2 ± √5
The discriminant is 16 + 4 = 20.
(4 ± 2√5) / 2 = 2 ± √5.
M1 — The discriminant is 16 + 4 = 20.
A1 — (4 ± 2√5) / 2 = 2 ± √5.
Question 3 Medium2 marks
Apply the quadratic formula to x² - 6x - 1 = 0 and simplify the exact roots.
Show answer
Answer: 3 ± √10
Discriminant 36 + 4 = 40 = 4 × 10.
(6 ± 2√10) / 2 = 3 ± √10.
M1 — Discriminant 36 + 4 = 40 = 4 × 10.
A1 — (6 ± 2√10) / 2 = 3 ± √10.
Question 4 Medium2 marks
Solve 2x² - 3x - 1 = 0 with the quadratic formula. a is not 1.
Show answer
Answer: (3 ± √17)/4
Denominator 2a = 4, discriminant 9 + 8 = 17.
The roots are (3 ± √17) / 4.
M1 — Denominator 2a = 4, discriminant 9 + 8 = 17.
A1 — The roots are (3 ± √17) / 4.
Question 5 Hard2 marks
Use the quadratic formula on 3x² - 2x - 2 = 0 and simplify the exact roots.
Show answer
Answer: (1 ± √7)/3
Discriminant 4 + 24 = 28 = 4 × 7.
(2 ± 2√7) / 6 = (1 ± √7) / 3.
M1 — Discriminant 4 + 24 = 28 = 4 × 7.
A1 — (2 ± 2√7) / 6 = (1 ± √7) / 3.
Question 6 Hard2 marks
x² - 8x + 2 = 0. Give both exact roots from the quadratic formula.
Show answer
Answer: 4 ± √14
Discriminant 64 - 8 = 56 = 4 × 14.
(8 ± 2√14) / 2 = 4 ± √14.
M1 — Discriminant 64 - 8 = 56 = 4 × 14.
A1 — (8 ± 2√14) / 2 = 4 ± √14.
Question 7 Exam-style4 marks
Solve 2x² + 5x - 1 = 0. Give your solutions correct to 3 significant figures.
Show answer
Answer: 0.186 and -2.69
a = 2, b = 5, c = -1.
Discriminant 25 + 8 = 33, and the denominator is 2a = 4.
x = (-5 ± √33) / 4.
x = 0.186 or x = -2.69.
M1 — Substituted a = 2, b = 5, c = -1 into the formula
A1 — (-5 ± √33) / 4
M1 — Evaluated either root
A1 — 0.186 and -2.69
Common mistakes
Most lost marks on this topic are one slip.
Rounding a surd root to 3 significant figures when the question asks for exact
Keep (-b ± √d) / (2a) and simplify the surd
Dividing by 2 when a is not 1
The denominator is 2a
Using b as positive when the term is -2x
The sign is part of b
Spot the slip
Sam solves x² - 2x - 1 = 0 and writes 2.41 and -0.41. What should he have kept?
The exact form 1 ± √2
Only the positive decimal
The discriminant on its own
Show the slip
The exact form 1 ± √2 The discriminant is 8, which simplifies to 1 ± √2, not a rounded decimal.
Exam tips
A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Three lines earn three chances at method marks: a,b,c; the discriminant; the substituted formula.
'To 2 decimal places' almost always means the formula, not factorising.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.