Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly. Get real or complex roots, discriminant, vertex, and axis of symmetry.
Quadratic Equation Solver
Solve ax² + bx + c = 0
How to Use
- 1 Enter the coefficients a, b, and c for your equation ax² + bx + c = 0
- 2 The solver instantly calculates the discriminant (b² − 4ac)
- 3 View the roots (x₁ and x₂), which may be real or complex numbers
- 4 Check the vertex coordinates and axis of symmetry for graphing
- 5 Use the Copy button to copy the full solution to your clipboard
What You Get
Quadratic equation solver that finds real and complex roots using the quadratic formula. Calculates the discriminant, vertex (h, k), axis of symmetry, and shows the nature of roots. Includes preset examples for common equation types.
Input: x² − 5x + 6 = 0 (a=1, b=−5, c=6)
Output: x₁ = 3, x₂ = 2 (Discriminant = 1, Vertex = (2.5, −0.25))
Input: x² + x + 1 = 0 (a=1, b=1, c=1)
Output: x₁ = −0.5 + 0.866i, x₂ = −0.5 − 0.866i (Complex roots)
Input: 2x² + 3x − 2 = 0 (a=2, b=3, c=−2)
Output: x₁ = 0.5, x₂ = −2 (Discriminant = 25, Vertex = (−0.75, −3.125))
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0, where a ≠ 0. The ± symbol means you compute two values: one with + and one with −, giving you the two roots x₁ and x₂.
What is the discriminant and what does it tell you?
The discriminant is the expression b² − 4ac under the square root in the quadratic formula. It determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one repeated real root; if Δ < 0, there are two complex conjugate roots (no real solutions).
How do you find the vertex of a parabola?
The vertex of a parabola y = ax² + bx + c has coordinates (h, k) where h = −b/(2a) and k = c − b²/(4a). The vertex is the minimum point if a > 0 (parabola opens upward) or the maximum point if a < 0 (parabola opens downward). It lies on the axis of symmetry x = h.
Can a quadratic equation have no solutions?
In the real number system, a quadratic equation has no real solutions when the discriminant (b² − 4ac) is negative. However, in the complex number system, every quadratic equation always has exactly two roots (which may be complex conjugates). This solver shows both real and complex roots.
What happens when a = 0 in ax² + bx + c = 0?
When a = 0, the equation is no longer quadratic — it becomes a linear equation bx + c = 0, with a single solution x = −c/b (assuming b ≠ 0). This solver requires a ≠ 0 to solve a true quadratic equation. For a = 0, simply solve the resulting linear equation.
How to solve x² − 5x + 6 = 0 step by step?
Identify a = 1, b = −5, c = 6. Calculate the discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1. Since Δ > 0, there are two real roots. Apply the quadratic formula: x = (5 ± √1) / 2 = (5 ± 1) / 2. So x₁ = (5+1)/2 = 3 and x₂ = (5−1)/2 = 2. You can verify by factoring: (x−3)(x−2) = 0.
What is the axis of symmetry of a quadratic equation?
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is x = −b/(2a). For example, for x² − 6x + 5 = 0, the axis of symmetry is x = 6/2 = 3.
How do complex roots work in quadratic equations?
Complex roots occur when the discriminant is negative. They always come in conjugate pairs: x = (−b + i√|Δ|)/(2a) and x = (−b − i√|Δ|)/(2a), where i = √(−1). For example, x² + 4 = 0 gives x = ±2i. Complex roots mean the parabola doesn't cross the x-axis.
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