Quadratic Equation Solver
Solve any quadratic equation using the quadratic formula. Shows real and complex roots, discriminant, vertex, and axis of symmetry.
Quadratic Solver
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Root x₂
2
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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
The Formula
The quadratic formula finds the roots (x-intercepts) of any quadratic equation ax² + bx + c = 0. The discriminant (b² − 4ac) determines the nature of the roots. The calculator also computes the vertex and axis of symmetry for the corresponding parabola.
Variable Definitions
Leading Coefficient
The coefficient of x². Must not be zero — if a = 0, the equation is linear, not quadratic. Controls the parabola's opening direction (up if a > 0, down if a < 0).
Linear Coefficient
The coefficient of x. Influences the position of the vertex along the x-axis.
Constant
The constant term. Controls where the parabola crosses the y-axis (the y-intercept at (0, c)).
b²−4ac
The key to root classification: positive = two distinct real roots, zero = one repeated root, negative = two complex conjugate roots.
How to Use This Calculator
- 1
Enter the coefficients a, b, and c from your quadratic equation in standard form: ax² + bx + c = 0.
- 2
The discriminant determines whether the roots are real or complex — it is displayed before the roots.
- 3
View the two solutions (roots), whether they are real or complex numbers.
- 4
The vertex and axis of symmetry are displayed to help understand the parabola geometry.
- 5
If a = 0, the calculator returns no results because the equation is not quadratic.
The quadratic formula finds the x-intercepts (roots) of a parabola defined by ax^2 + bx + c = 0
Understanding the Concept
The quadratic formula is the universal method for solving any quadratic equation, derived by completing the square on the general form ax² + bx + c = 0. It has been known since ancient Babylonian mathematicians (circa 2000 BCE) and was fully formalized by al-Khwarizmi in the 9th century. The discriminant (b² − 4ac) tells you the nature of the solutions before calculating them. A positive discriminant means two distinct real roots — the parabola crosses the x-axis at two points. A zero discriminant means one repeated root (a double root) — the parabola just touches the x-axis at its vertex. A negative discriminant means no real roots — the solutions are complex conjugates in the form (p ± qi), where i² = −1, and the parabola does not cross the x-axis at all. The vertex (h, k) is the parabola's minimum (if a > 0) or maximum (if a < 0) point, occurring at h = −b/2a, k = a·h² + b·h + c. The axis of symmetry is the vertical line x = h through the vertex. Quadratics appear everywhere: projectile motion (height = −½gt² + v₀t + h₀), area optimization problems, structural engineering (parabolic arches), and economics (profit functions).
Frequently Asked Questions
Sources & References
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