Quadratic Equation Solver

Solve quadratic equations ax² + bx + c = 0 for real or complex conjugate roots, discriminant (Δ = b² - 4ac), Vieta sum/product, and parabola vertex (h, k).

Category: Math

How to Use Quadratic Equation Solver

  1. Enter the quadratic coefficient a (where a ≠ 0), linear coefficient b, and constant term c for ax² + bx + c = 0.
  2. Click Calculate to evaluate the discriminant Δ = b² − 4ac and apply the quadratic formula.
  3. Review roots x₁ and x₂ (real or complex conjugate), sum and product of roots, parabola vertex (h, k), axis of symmetry, and y-intercept.

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Frequently Asked Questions

What is the quadratic formula used to solve ax² + bx + c = 0?

For any quadratic equation ax² + bx + c = 0 where a ≠ 0, the roots are given by x = (−b ± √(b² − 4ac)) / (2a). The solver computes both x₁ and x₂ whether coefficients b or c are positive, negative, decimal, or zero.

What does the Discriminant (Δ = b² − 4ac) tell you about the roots?

The discriminant Δ = b² − 4ac determines the nature of the roots: 1) If Δ > 0, the equation has two distinct real roots (the parabola crosses the x-axis twice); 2) If Δ = 0, it has one repeated real root x = −b / (2a) (the vertex touches the x-axis); 3) If Δ < 0, it has two complex conjugate roots (−b/(2a) ± i√|Δ|/|2a|).

How are the Parabola Vertex (h, k) and Axis of Symmetry calculated?

The graph of y = ax² + bx + c is a parabola opening upward when a > 0 and downward when a < 0. Its axis of symmetry is the vertical line x = h = −b / (2a), and its vertex minimum or maximum point is located at (h, k) where k = c − b² / (4a).

What are Vieta's formulas for the sum and product of quadratic roots?

By Vieta's formulas, for any quadratic equation ax² + bx + c = 0 (even when the roots are complex), the sum of the roots x₁ + x₂ always equals −b / a, and the product of the roots x₁ × x₂ always equals c / a.

Why cannot coefficient 'a' be equal to zero?

If a = 0, the x² term vanishes and ax² + bx + c = 0 reduces to the linear equation bx + c = 0 (whose single root is x = −c / b). The quadratic formula divides by 2a, which would cause division by zero when a = 0.