How do you solve a quadratic equation using the quadratic formula?
Short answer
Write the equation as ax² + bx + c = 0, read off a, b and c, then substitute them into x = (−b ± √(b² − 4ac)) / 2a. Work out the discriminant b² − 4ac first: if it is positive there are two solutions, if it is zero there is one, and if it is negative there are no real solutions.
Step by step
- Rearrange the equation so one side is zero: ax² + bx + c = 0.
- Write down a, b and c, keeping their signs. In x² − 3x − 10 = 0, a = 1, b = −3 and c = −10.
- Calculate the discriminant b² − 4ac on its own first. This avoids most sign errors.
- Substitute into x = (−b ± √(b² − 4ac)) / 2a and work out the + and − versions separately.
- Check each answer by putting it back into the original equation.
Worked example
Solve 2x² + 5x − 3 = 0
- a = 2, b = 5, c = −3
- Discriminant: 5² − 4 × 2 × (−3) = 25 + 24 = 49
- x = (−5 ± √49) / 4 = (−5 ± 7) / 4
- x = 2/4 = 0.5 or x = −12/4 = −3
Common mistakes
- Forgetting that b is negative when the equation has −bx, so −b becomes positive.
- Dividing only the square root by 2a instead of the whole top line.
- Squaring a negative b on the calculator without brackets: type (−3)², not −3².
Useful for:GCSEIGCSEWAECMatricBac
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