2x² − 4x − 6 = 0
D = (−4)² − 4 × 2 × (−6) = 64. x = (4 ± 8) ÷ 4: 3 and −1. Vertex (1, −8).
Quadratic equation solver with steps
Two real roots
3, -1
Vertex at (1, -8)
How we got this
- Discriminant D = b² − 4ac = (-4)² − 4 × 2 × (-6) = 64.
- D > 0, so there are two real roots.
- x = (−b ± √D) ÷ 2a = (4 ± 8) ÷ 4.
- Answer: x = 3, -1.
Discriminant first
Work out b² − 4ac before anything else. Its sign tells you what kind of answer to expect and whether factoring is worth trying: a perfect-square discriminant means the roots are rational and the equation factors. Then put it into the formula, keeping brackets around negative values so the signs survive.
The square root step is where most marks are lost; the square root calculator gives it in exact form. Fractional roots simplify with the fraction calculator.
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What this calculator does not account for
- Complex roots are shown, but not graphed.
- Equations of a higher degree.
How this is calculated
Formula: Quadratic formula
x =(−b ± √(b² − 4ac)) ÷ 2aWhat the symbols mean
- D = b² − 4acThe discriminant. Positive: two real roots. Zero: one. Negative: two complex roots.
- vertexThe turning point, at x = −b ÷ 2a.
Good to know
- If a = 0 the equation is linear, not quadratic.
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Worked examples
Each example is worked out by the same code as the calculator above.
x² + 2x + 5 = 0
D = 4 − 20 = −16. x = −1 ± 2i.
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<iframe src="/embed/quadratic-equation" title="Quadratic solver" width="100%" height="720" style="border:0;max-width:42rem" loading="lazy"></iframe>Questions
How do I solve 2x² − 4x − 6 = 0?
D = 16 + 48 = 64, √D = 8. x = (4 ± 8) ÷ 4, so x = 3 or x = −1.
What does a negative discriminant mean?
The parabola never crosses the x-axis. The roots are complex: −b ÷ 2a ± i√(−D) ÷ 2a.
Should I factor or use the formula?
Factor when the numbers are small and the roots are whole. The formula always works.