How a Quadratic Equation Solver Works
A quadratic equation has the form ax² + bx + c = 0, and this solver finds its roots using the quadratic formula. Enter the three coefficients a, b, and c, and the calculator returns the roots, the discriminant, the vertex of the parabola, and its axis of symmetry, with the formula filled in using your numbers.
The discriminant, b² − 4ac, tells you in advance what kind of roots to expect. When it is negative, the roots are complex, and the calculator formats them neatly as a conjugate pair using the imaginary unit i.
The Formulas
- Roots: x = (−b ± √(b² − 4ac)) / (2a)
- Discriminant: Δ = b² − 4ac
- Vertex: (−b / 2a, c − b² / 4a)
- Axis of symmetry: x = −b / 2a
Understanding the Parabola
Every quadratic graphs as a parabola, and the coefficients control its shape and position. The sign of a decides whether it opens up or down, and the vertex marks its highest or lowest point.
- If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.
- The roots are the x-values where the parabola crosses the x-axis.
- The axis of symmetry is the vertical line through the vertex that mirrors the two halves of the curve.
- A discriminant of zero means the vertex sits exactly on the x-axis, giving one repeated root.
Frequently Asked Questions
What does the discriminant tell me?
The discriminant b² − 4ac reveals the nature of the roots before you solve. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex conjugate roots.
Why must a not equal zero?
If a is zero, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula divides by 2a, so a must be non-zero for it to apply.
How are complex roots written?
When the discriminant is negative, the roots are a complex conjugate pair of the form p ± qi, where p is the real part (−b/2a) and q is the imaginary part (√|Δ| / 2a). The calculator displays both roots in this format automatically.