Quadratic Formula Calculator
Solve quadratic equations ax²+bx+c=0 using the quadratic formula. Find roots, discriminant, and vertex coordinates instantly.
Root x₁
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Root x₂ —
Discriminant (Δ) —
Vertex X —
Vertex Y —
Root Type —
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Root x₁
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Root x₂ —
Discriminant (Δ) —
Root Type —
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Roots
x₁ (Real Part) —
x₁ (Imaginary Part) —
x₂ (Real Part) —
x₂ (Imaginary Part) —
Factored / Complex Form —
Discriminant
Discriminant (Δ = b²−4ac) —
Discriminant Analysis —
Graph Properties
Vertex X —
Vertex Y —
Axis of Symmetry x = —
Vieta's Formulas
Sum of Roots (−b/a) —
Product of Roots (c/a) —
How to Use This Calculator
Enter coefficients a, b, and c for the equation ax²+bx+c=0. The calculator shows both roots (if real), the discriminant, and the vertex of the parabola.
Formula
x = (−b ± √(b²−4ac)) / 2a • Vertex: (−b/2a, f(−b/2a))
Example
x²−5x+6=0: Δ=1 → x=3, x=2 • Vertex at x=2.5, y=−0.25
Frequently Asked Questions
- The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / (2a). The ± sign means there are generally two solutions. For example, to solve x² − 5x + 6 = 0 (a=1, b=−5, c=6): x = (5 ± √(25−24)) / 2 = (5 ± 1) / 2, giving x = 3 and x = 2. You can verify: (x−3)(x−2) = x² − 5x + 6. The formula works for any quadratic regardless of whether the equation factors nicely.
- The discriminant Δ = b² − 4ac determines the nature of the roots without solving the equation. If Δ > 0: two distinct real roots (the parabola crosses the x-axis twice). If Δ = 0: exactly one repeated real root (the parabola just touches the x-axis — called a double root). If Δ < 0: two complex conjugate roots (the parabola does not cross the x-axis at all). For example, x² + 1 = 0 has Δ = 0 − 4 = −4 < 0, so its roots are the complex numbers x = ±i.
- The vertex is the turning point of the parabola — the minimum if a > 0 or the maximum if a < 0. Its x-coordinate is x_v = −b/(2a), and the y-coordinate is found by substituting back: y_v = a(x_v)² + b(x_v) + c. Example: for x² − 6x + 8 = 0, vertex x = −(−6)/(2×1) = 3 and y = 9 − 18 + 8 = −1. So the vertex is (3, −1). The axis of symmetry passes through x_v, meaning the two roots are symmetric around that line.
- The coefficient a controls both the direction and width of the parabola. If a > 0, the parabola opens upward and has a minimum. If a < 0, it opens downward and has a maximum. The larger |a| is, the narrower and steeper the parabola; the smaller |a|, the wider and flatter. For example, y = 10x² is a very narrow upward parabola, while y = 0.1x² is very wide. Multiplying a by 4 makes the parabola four times narrower.
- No — if a = 0, the equation reduces to bx + c = 0, which is linear, not quadratic. The quadratic formula would have division by zero (2a = 0), so it cannot be applied. Always ensure a ≠ 0 before using this calculator. If a = 0, solve bx + c = 0 directly: x = −c/b (assuming b ≠ 0). If both a and b are zero, the equation becomes c = 0, which is either always true (c = 0) or has no solution (c ≠ 0).