Quadratic Solver
Roots via quadratic formula
ax² + bx + c = 0
About This Calculator
A quadratic equation is a polynomial of degree 2 in the form ax² + bx + c = 0. Its solutions (roots) can be found using the quadratic formula, completing the square, or factoring. The discriminant (b²−4ac) determines whether the roots are real, repeated, or complex.
Formula
ax² + bx + c = 0
x = (−b ± √(b²−4ac)) / (2a)
Discriminant D = b²−4ac: D>0 → 2 real roots; D=0 → 1 repeated root; D<0 → 2 complex roots
Sum of roots = −b/a; Product of roots = c/a
Example Calculation
Solve 2x² − 4x − 6 = 0.
- a=2, b=−4, c=−6
- D = (−4)² − 4(2)(−6) = 16 + 48 = 64
- x = (4 ± √64) / 4 = (4 ± 8) / 4
- x₁ = 12/4 = 3; x₂ = −4/4 = −1
x = 3 or x = −1
Discriminant and Root Types
| D = b²−4ac | Root Type | Example equation | Roots |
|---|---|---|---|
| D > 0 | Two distinct real roots | x²−5x+6=0 | x=2, x=3 |
| D = 0 | One repeated real root | x²−4x+4=0 | x=2 (double) |
| D < 0 | Two complex conjugate roots | x²+1=0 | x=±i |
Frequently Asked Questions
What does the discriminant tell me?
The discriminant b²−4ac determines the nature of the roots without solving the equation. Positive means two distinct real roots, zero means one repeated root, and negative means complex (imaginary) roots.
Can I always use the quadratic formula?
Yes — the quadratic formula always works for any quadratic equation with real coefficients. Factoring is faster when integer roots exist, but the formula handles all cases including complex roots.
What is completing the square?
Completing the square rewrites ax²+bx+c as a(x+h)²+k, revealing the vertex of the parabola. It is the method used to derive the quadratic formula and is also useful for converting conic section equations.
What does the graph of a quadratic look like?
A quadratic function graphs as a parabola. If a>0 it opens upward (U-shape); if a<0 it opens downward. The vertex is the minimum or maximum point, and the roots are where the parabola crosses the x-axis.