Quadratic equation solver
A quadratic equation ax² + bx + c = 0 is solved through its discriminant D = b² − 4ac: a positive D gives two real roots, a D of zero gives one repeated root, and a negative D gives a pair of complex conjugate roots.
- x₂
- 2
- Number of roots
- Two real roots (D > 0)
- Discriminant D
- 1
- Vertex x
- 2.5
Rates and thresholds
- Root formula
- x = (−b ± √D) / 2a
- Discriminant
- D = b² − 4ac
- Sum of roots
- x₁ + x₂ = −b / a
- Product of roots
- x₁ · x₂ = c / a
- Vertex of the parabola
- x = −b / 2a
What does the discriminant tell you?
The discriminant D = b² − 4ac tells you how many roots the equation has without your having to find them. D > 0 means two distinct real roots, D = 0 means one repeated root, and D < 0 means two complex conjugate roots. Graphically: with D > 0 the parabola crosses the x-axis twice, with D = 0 it touches it once, and with D < 0 it misses it entirely.
What if a = 0?
If a = 0 the equation is not quadratic but linear, bx + c = 0, with the single solution x = −c / b. The root formula cannot be used because it would divide by zero. If b is also zero, the equation either has no solution (c ≠ 0) or holds for every x (c = 0).
What are Vieta's formulas and when do they help?
Vieta's formulas say that the roots of a quadratic sum to −b / a and multiply to c / a. They often let you find the roots in your head: for x² − 5x + 6 = 0 the roots sum to 5 and multiply to 6, so they are 2 and 3. They are just as useful for checking an answer — if your roots do not satisfy both, the arithmetic went wrong somewhere.
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