Quadratic Equation Solver
Solve quadratic equations with step-by-step solutions using the quadratic formula. Shows discriminant, both roots (real or complex), vertex, axis of symmetry, and an interactive parabola graph.
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How to Use
- 1 Enter coefficient a — Enter the coefficient of x² (must not be zero).
- 2 Enter coefficient b — Enter the coefficient of x.
- 3 Enter coefficient c — Enter the constant term.
- 4 Click Solve Equation — Press the Solve button to find the roots.
- 5 Review the complete solution — View both roots, discriminant analysis, step-by-step solution, parabola properties, and graph.
💡 Pro Tip: The discriminant (Δ = b² - 4ac) tells you about the roots before solving: positive = 2 real roots, zero = 1 repeated root, negative = 2 complex roots.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0. The "a" coefficient determines if the parabola opens upward (a > 0) or downward (a < 0). Common examples include x² - 5x + 6 = 0 and 2x² + 3x - 2 = 0.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b²-4ac)) / 2a. This formula gives both solutions (roots) of any quadratic equation. The ± means you calculate twice: once with + and once with - to get both roots.
What is the discriminant?
The discriminant is Δ = b² - 4ac. It determines the nature of the roots: If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root (double root). If Δ < 0, there are two complex conjugate roots.
What are complex roots?
When the discriminant is negative, the equation has no real solutions but has two complex conjugate roots in the form a ± bi, where i = √(-1). For example, x² + 1 = 0 has roots x = ±i.
What does the parabola graph show?
The graph shows the parabola y = ax² + bx + c. The x-intercepts (where the curve crosses the x-axis) are the real roots of the equation. The vertex is the highest or lowest point, and the axis of symmetry passes through it.