Quadratic Equation Solver

A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a, b, and c are real coefficients and a ≠ 0. The term "quadratic" comes from the Latin quadratus, meaning square, referring to the x² term. Quadratic equations appear throughout mathematics, physics, engineering, and finance — any time a relationship involves a squared variable. Our quadratic equation solver handles all three root cases: two distinct real roots, one repeated real root, and two complex conjugate roots, determined entirely by the value of the discriminant (Δ = b² − 4ac).

1. 1. Enter the Coefficients: Input the values of a, b, and c for your equation ax² + bx + c = 0. The equation preview updates in real time so you can confirm the equation before solving. All processing happens locally in your browser — your data never leaves your device.
2. 2. Instant Browser-Based Solving: Click "Solve Equation" and the solver computes the discriminant, applies the quadratic formula, and classifies the roots. It also calculates the vertex, axis of symmetry, y-intercept, and Vieta's formulas in the same pass.
3. 3. Review Results and Steps: View the roots with copy buttons, the full properties grid, and toggle the step-by-step solution to see every stage of the quadratic formula applied to your specific equation.

• Discriminant (Δ): Δ = b² − 4ac determines the nature of the roots. Positive Δ gives two real roots, zero Δ gives one repeated root, and negative Δ gives two complex conjugate roots.
• Roots (x₁ and x₂): Computed using the quadratic formula x = (−b ± √Δ) / 2a. Complex roots are displayed in standard a + bi form with the real and imaginary parts separated.
• Vertex and Axis of Symmetry: The vertex of the parabola y = ax² + bx + c is at (−b/2a, f(−b/2a)). The axis of symmetry is the vertical line x = −b/2a, which passes through the vertex.
• Vieta's Formulas: The sum of the roots equals −b/a and the product of the roots equals c/a, regardless of whether the roots are real or complex. These relationships are useful for verification and for constructing equations from known roots.

When the discriminant is negative, the quadratic equation has no real solutions — the parabola does not intersect the x-axis. The two roots are complex conjugates of the form p + qi and p − qi, where p = −b/2a and q = √(−Δ)/2a. Complex roots always come in conjugate pairs when the coefficients a, b, and c are real numbers. While complex roots cannot be plotted on a standard number line, they are essential in electrical engineering (impedance), control systems (stability analysis), and quantum mechanics.