The quadratic formula solves any equation of the form ax² + bx + c = 0, even when it can't be factored. You just plug in a, b and c.
Try it with your own problem ↓
The steps
- Move everything to one side so the equation equals 0, and read off a, b and c.
- Write the formula: x = (−b ± √(b² − 4ac)) ÷ (2a).
- Work out the discriminant, b² − 4ac, first. If it is positive there are two real solutions, if it is 0 there is one, and if it is negative there are no real solutions.
- Take the square root and work out the "+" and "−" versions separately.
- Simplify each answer, and check by putting it back into the equation.
A worked example
Solve 2x² − 5x − 3 = 0.
- Here a = 2, b = −5 and c = −3.
- Discriminant: b² − 4ac = (−5)² − 4(2)(−3) = 25 + 24 = 49. Its square root is 7.
- x = (−(−5) ± 7) ÷ (2 · 2) = (5 ± 7) ÷ 4.
- Plus: (5 + 7) ÷ 4 = 12 ÷ 4 = 3.
- Minus: (5 − 7) ÷ 4 = −2 ÷ 4 = −1/2.
- Check x = 3: 2(9) − 15 − 3 = 0. ✓
x = 3 or x = −1/2.
Common mistakes
- Forgetting the sign of b. If b = −5 then −b = 5, and b² = 25.
- Dividing only part of the numerator by 2a. The whole top is divided by 2a.
- Calculating b² − 4ac wrong. Do (−5)² = 25 and then subtract 4ac. A negative c makes −4ac positive.
- Forgetting the ±. There are two answers unless the discriminant is 0.
Questions people ask
What does the discriminant tell me?
It tells how many real solutions there are: positive means two, zero means one, and negative means none (the solutions are complex numbers).
When should I use the formula rather than factoring?
When the quadratic doesn't factor easily. It always works.
Where does the formula come from?
From completing the square on ax² + bx + c = 0 in general.
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