Quadratic Functions and Equations in One Variable
How to Solve using the quadratic formula
Use the quadratic formula when a quadratic equation does not factorise and you need the roots exactly.
Before you start
- Rearranging an equation into ax² + bx + c = 0
- Substituting negative numbers using brackets
- Finding a square root on your calculator
When to use it
Use the quadratic formula when a quadratic equation does not factorise and you need the roots exactly.
The steps
- Write the equation in the form ax² + bx + c = 0 and read off a, b and c with their signs.
- Compute the discriminant b² − 4ac first, it tells you how many roots to expect.
- Substitute a, b and c into x = (−b ± √(b²−4ac)) / (2a).
- Work out the two values, keeping the ± as two separate cases.
- Round only at the final step, to the accuracy the question asks.
Worked example
Solve x² + 5x + 3 = 0, giving your answers correct to 2 decimal places.
- Compare with ax² + bx + c = 0: a = 1, b = 5, c = 3.
- Discriminant b² − 4ac = 5² − 4(1)(3) = 25 − 12 = 13, which is positive, so there are two real roots.
- Substitute into x = (−b ± √(b²−4ac)) / (2a) = (−5 ± √13) / 2.
- Two cases: x = (−5 + 3.6056)/2 = −0.6972 and x = (−5 − 3.6056)/2 = −4.3028.
- Round at the final step to 2 decimal places: x ≈ −0.70 or x ≈ −4.30.
A second example, with a twist
Here a is not 1 and c is negative, so the signs inside b² − 4ac need extra care. Solve 2x² − 4x − 3 = 0, giving your answers correct to 2 decimal places.
- Compare with ax² + bx + c = 0: a = 2, b = −4, c = −3.
- Discriminant b² − 4ac = (−4)² − 4(2)(−3) = 16 + 24 = 40, positive, so two real roots.
- Substitute: x = (−(−4) ± √40) / (2×2) = (4 ± √40) / 4.
- Two cases: x = (4 + 6.3246)/4 = 2.5811 and x = (4 − 6.3246)/4 = −0.5811.
- Round at the final step to 2 decimal places: x ≈ 2.58 or x ≈ −0.58.
Practise this in a KBAT problem
Frequently asked questions
How do I know how many roots there are before solving?
Work out the discriminant b² − 4ac first. If it is positive there are two different real roots, if it is exactly zero there is one repeated root, and if it is negative there are no real roots.
Checking it early tells you what kind of answer to expect.
Should I round the square root partway through?
No, round only at the very end. If you round √13 or √40 too early, the small error grows through the division and your final answer can be off.
Keep the full calculator value while you work, and round to the accuracy the question asks only on the last line.
What does the ± sign actually mean here?
The ± tells you to do the calculation twice: once with a plus and once with a minus. Each choice gives one root, so a quadratic usually has two answers.
Write them out as two separate cases so you do not accidentally lose one of the roots.