Maths
Quadratic formula — practice & summary
The quadratic formula solves any equation of the form ax² + bx + c = 0 (a ≠ 0). The solutions are x = (-b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is the discriminant — it tells you how many real roots exist before you even solve.
Key points
- Formula: x = (−b ± √(b² − 4ac)) / 2a
- Discriminant > 0 → two distinct real roots
- Discriminant = 0 → one repeated real root
- Discriminant < 0 → no real roots (two complex conjugate roots)
- Always check by substitution if the question asks for verification
In more depth
Using the formula step by step
Rearrange the equation to ax² + bx + c = 0 and identify a, b and c, keeping their signs. Work out the discriminant b² − 4ac first. Then substitute into x = (−b ± √(b² − 4ac)) / 2a, calculating the + and − versions separately. Check each answer by substituting it back into the original equation.
What the discriminant tells you
If b² − 4ac is positive, there are two different real roots and the parabola crosses the x-axis twice. If it equals zero, there is one repeated root and the graph just touches the axis. If it is negative, there are no real roots and the graph stays entirely above or below the axis.
When to use another method
Factorising is quicker when the quadratic factors neatly, and completing the square reveals the turning point of the graph. The formula always works, so it is the safe choice when other methods do not come easily.
Common mistakes
- Dividing only −b by 2a instead of the whole numerator.
- Losing negative signs: when b = −5, −b is +5 and b² is +25.
- Forgetting to rearrange the equation to equal zero before reading off a, b and c.
Practice quiz
Click each question to reveal the answer.
1. Solve x² − 5x + 6 = 0
Answer: x = 2 or x = 3
Factorises to (x−2)(x−3) = 0, so x = 2 or 3.
2. What is the discriminant of 2x² + 3x − 5?
Answer: 49
b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49.
3. How many real roots does x² + x + 1 = 0 have?
- 0
- 1
- 2
- Infinitely many
Answer: 0
Discriminant = 1 − 4 = −3 < 0, so no real roots.
Last reviewed: May 2026