Quadratic Equation Solver
Solve any quadratic equation ax²+bx+c=0 instantly. Enter the three coefficients and get real or complex roots, the discriminant, vertex, axis of symmetry, and a full step-by-step working.
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Enter Coefficients
Fill in a, b, and c for the equation ax² + bx + c = 0. Coefficient a must not be zero.
Results
Equation & Roots
| Equation | — |
|---|---|
| Discriminant (b² − 4ac) | — |
| Root x₁ | — |
| Root x₂ | — |
Parabola Properties
| Vertex x | — |
|---|---|
| Vertex y | — |
| Axis of Symmetry | — |
| Parabola Opens | — |
Step-by-Step Working
Understanding Quadratic Equations
The Quadratic Formula
Any equation of the form ax²+bx+c=0 (with a ≠ 0) can be solved using the quadratic formula:
x = (−b ± √(b²−4ac)) / (2a)
The ± sign means the formula produces two solutions simultaneously — one with addition and one with subtraction. Together they are called x₁ and x₂.
- a — controls the width and direction of the parabola.
- b — shifts the parabola horizontally.
- c — is the y-intercept of the parabola.
The Discriminant Explained
The discriminant Δ = b²−4ac is the expression under the square root in the quadratic formula. It tells you the nature of the roots before you solve:
- Δ > 0 — two distinct real roots. The parabola crosses the x-axis at two points.
- Δ = 0 — one repeated real root. The parabola just touches the x-axis at its vertex.
- Δ < 0 — two complex conjugate roots. The parabola does not cross the x-axis.
Real vs Complex Roots
Real roots are ordinary numbers — they appear on the number line and represent where the parabola crosses the x-axis.
Complex roots involve the imaginary unit i (where i² = −1). They come in conjugate pairs: x = p+qi and x = p−qi. Although they are not visible on the real number line, they are valid solutions of the equation and important in physics and engineering.
Example: x²+1=0 has no real solutions but has complex roots x = ±i.
Vertex and Parabola Shape
The graph of ax²+bx+c is a parabola. Its key features are:
- Vertex — the turning point at x = −b/(2a), y = c − b²/(4a).
- Axis of symmetry — the vertical line x = −b/(2a) that divides the parabola into two mirror halves.
- Opens upward when a > 0 (vertex is a minimum).
- Opens downward when a < 0 (vertex is a maximum).
The vertex x-coordinate is also the average of the two real roots (when they exist).
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What is the quadratic formula?
The quadratic formula solves any equation of the form ax²+bx+c=0. The formula is x = (−b ± √(b²−4ac)) / (2a). The ± gives two solutions simultaneously. It works for all real values of a, b, and c as long as a ≠ 0.
What is the discriminant and what does it tell you?
The discriminant is b²−4ac. If it is positive, the equation has two distinct real roots. If it equals zero, there is exactly one real (repeated) root. If it is negative, the equation has two complex conjugate roots. The discriminant reveals the nature of the solutions without fully solving the equation.
What are complex roots?
When the discriminant is negative, the square root of a negative number appears in the formula, producing complex roots. They take the form x = p ± qi, where p = −b/(2a) and q = √(−disc)/(2a). Complex roots always come in conjugate pairs and do not correspond to real x-intercepts on the parabola.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola y = ax²+bx+c. Its coordinates are x = −b/(2a) and y = c − b²/(4a). The parabola opens upward when a > 0 (vertex is the minimum) and downward when a < 0 (vertex is the maximum).
What is the difference between a quadratic and a linear equation?
A linear equation (ax+b=0, degree 1) has exactly one solution and graphs as a straight line. A quadratic equation (ax²+bx+c=0, degree 2) can have zero, one, or two solutions and graphs as a parabola. Quadratics require the quadratic formula (or factoring/completing the square) to solve, whereas linear equations are solved by simple rearrangement.