Quadratic Equation Solver — Find Roots, Vertex & Discriminant
Solve any quadratic ax²+bx+c=0 with the quadratic formula. Get real or complex roots, discriminant, vertex, axis of symmetry, factored form, and Vieta.
Quadratic Solver
Results below
Enter Values
Discriminant (Δ = b² − 4ac)
▲ 1
Vertex (h, k)
(2.5, -0.25)
Axis of Symmetry
x = 2.5
Y-intercept
(0, 6)
Parabola Opens
Upward (∪) — minimum at vertex
Root Nature
▲ Two distinct real roots (Δ > 0)
Root x₁
▲ 3
Root x₂
▲ 2
Sum of Roots (Vieta)
5
Product of Roots (Vieta)
6
Factored Form
▲ (x − 3)(x − 2)
Scenario Comparison
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Equation
1x² - 5x + 6 = 0
Parabola f(x) = ax² + bx + c
Step-by-Step Solution
Step 1: Identify Coefficients
a = 1, b = -5, c = 6
Step 2: Calculate the Discriminant
Δ = b² − 4ac = -5² − 4(1)(6)
Δ = 25 − 24 = 1
Step 3: Interpret the Discriminant
Δ = 1 — The discriminant is POSITIVE, so there are two distinct real roots. The parabola crosses the x-axis at two points.
Step 4: Apply the Quadratic Formula
x = (−b ± √Δ) / (2a)
x = (5 ± √1) / (2 × 1)
Step 5: Real Roots
x₁ = (−-5 + √1) / 2 = 3
x₂ = (−-5 − √1) / 2 = 2
Vertex
(2.5, -0.25)
Minimum point (opens upward)
Axis of Symmetry
x = 2.5
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