Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Shows real and complex roots, discriminant, vertex, and step-by-step solution.
ax² + bx + c = 0
Two Real Roots (Δ > 0)
1x² -5x +6 = 0
x1 = 3
x2 = 2
Discriminant (Δ)
1
Vertex x (axis of symmetry)
2.5
Vertex y (min/max)
-0.25
Parabola opens
Upward ∪ (minimum)
Sum of roots (−b/a)
5
Product of roots (c/a)
6
Step-by-Step Solution
1.
Equation: 1x² -5x +6 = 0
2.
Discriminant: Δ = b² − 4ac = -5² − 4(1)(6) = 25 − 24 = 1
3.
Δ > 0 → Two distinct real roots
4.
x = (−b ± √Δ) / 2a = (−(-5) ± √1) / (2 × 1)
5.
x = (5 ± 1.00000) / 2
6.
x₁ = 3.0000000, x₂ = 2.0000000
7.
Vertex: (2.50000, -0.250000)
8.
Axis of symmetry: x = 2.50000
How to Use Quadratic Equation Solver
- 1Enter coefficients a, b, and c for the equation ax² + bx + c = 0.
- 2See the discriminant, roots (real or complex), and vertex.
- 3View the step-by-step solution using the quadratic formula.
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Frequently Asked Questions
What is a quadratic equation?▾
A quadratic equation is a polynomial equation of degree 2: ax² + bx + c = 0, where a ≠ 0. The term "quadratic" comes from "quadratum" (Latin: square). Examples: x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, x² + 1 = 0. Real-world applications: projectile motion (height = −16t² + vt + h₀), profit maximization (revenue − cost curves), area problems (rectangle with given area), physics (acceleration, force). Every quadratic has exactly 2 roots (counting multiplicity), which may be real or complex.
How does the quadratic formula work?▾
For ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / (2a). The "±" gives two solutions: x₁ = (−b + √discriminant) / 2a and x₂ = (−b − √discriminant) / 2a. Derivation by completing the square: ax² + bx + c = 0 → x² + (b/a)x = −c/a → (x + b/2a)² = (b² − 4ac)/(4a²) → x = (−b ± √(b²−4ac))/(2a). Discovered independently by Al-Khwarizmi (~820 AD) and others. The discriminant b²−4ac determines the nature of roots.
What does the discriminant tell you?▾
Discriminant (Δ) = b² − 4ac. If Δ > 0: two distinct real roots. The parabola crosses the x-axis at two points. If Δ = 0: one repeated real root (double root). The parabola touches the x-axis at exactly one point (vertex). If Δ < 0: two complex conjugate roots (a ± bi). The parabola does not cross the x-axis. Example: x² − 5x + 6 = 0 → Δ = 25 − 24 = 1 > 0 → two real roots (2 and 3). x² − 2x + 1 = 0 → Δ = 4 − 4 = 0 → one root (1). x² + 1 = 0 → Δ = 0 − 4 = −4 → complex roots (±i).
What is the vertex of a parabola?▾
The vertex is the maximum or minimum point of the parabola y = ax² + bx + c. Vertex x-coordinate: h = −b/(2a) (axis of symmetry). Vertex y-coordinate: k = c − b²/(4a) = f(h). If a > 0: parabola opens upward, vertex is a minimum. If a < 0: parabola opens downward, vertex is a maximum. Vertex form: y = a(x − h)² + k. The vertex x-coordinate is also the average of the two roots: h = (x₁ + x₂)/2. By Vieta's formulas: x₁ + x₂ = −b/a and x₁ × x₂ = c/a.
What is Vieta's formulas?▾
For ax² + bx + c = 0 with roots x₁, x₂: Sum of roots: x₁ + x₂ = −b/a. Product of roots: x₁ × x₂ = c/a. These allow you to reconstruct the equation from its roots: ax² + bx + c = a(x − x₁)(x − x₂). Applications: quickly check your answers (sum and product should match). Factor quadratics mentally: for x² + bx + c = 0, find two numbers that add to b and multiply to c. Example: x² − 5x + 6 = 0 → need numbers summing to 5 and multiplying to 6 → 2 and 3. So x² − 5x + 6 = (x−2)(x−3).