Quadratic Equation Solver
Solve ax² + bx + c = 0 with real/complex roots, discriminant, parabolic vertex, and step-by-step formula derivations.
Quadratic Equation Solver (ax² + bx + c = 0)
Calculate discriminant, real/complex roots, parabolic vertex, and step-by-step formula working
Roots of the Equation
Nature: Two distinct real roots (Δ = 1)
How Does It Work?
Free client-side quadratic equation solver. Enter coefficients a, b, and c to calculate discriminant (Δ), real and complex conjugate roots, vertex coordinates (h, k), axis of symmetry, and full step-by-step quadratic formula working.
The quadratic formula yields solutions for any second-degree polynomial equation. The discriminant determines whether roots are real and distinct (Δ > 0), repeated real (Δ = 0), or complex conjugates (Δ < 0).
Real-World Worked Example
Solving the quadratic equation x² - 5x + 6 = 0 where a = 1, b = -5, and c = 6.
- •Identify coefficients: a = 1, b = -5, c = 6
- •Compute Discriminant: Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
- •Nature of Roots: Δ > 0 (Two distinct real roots)
- •Calculate Square Root of Δ: √1 = 1
- •Root 1 (x₁): [-(-5) + 1] / (2 × 1) = (5 + 1) / 2 = 6 / 2 = 3
- •Root 2 (x₂): [-(-5) - 1] / (2 × 1) = (5 - 1) / 2 = 4 / 2 = 2
- •Vertex Coordinates (h, k): h = -(-5) / (2 × 1) = 2.5; k = 6 - (-5)² / (4 × 1) = 6 - 6.25 = -0.25
- •Vertex Type: Minimum (since a = 1 > 0)
- •Axis of Symmetry: x = 2.5
Takeaway: The roots x = 3 and x = 2 represent the exact points where the parabola intersects the x-axis, and the vertex (2.5, -0.25) marks the lowest point of the curve.
Complete Guide & Practical Explanations
A quadratic equation is a second-order polynomial equation in a single variable x, written in the standard form: ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Quadratic relationships model numerous physical phenomena, from the parabolic trajectory of a thrown cricket ball to revenue optimization curves and projectile motion in engineering.
Every quadratic equation is solved using the universal quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). The term under the square root radical, b² - 4ac, is called the Discriminant (denoted by Δ). The discriminant reveals the nature of the roots before calculating the full formula:
1. Positive Discriminant (Δ > 0): The equation possesses two distinct real roots. The parabola crosses the x-axis at two separate locations. For instance, in x² - 5x + 6 = 0, Δ = 25 - 24 = 1 > 0, yielding real roots x = 3 and x = 2.
2. Zero Discriminant (Δ = 0): The equation has exactly one repeated real root (also called a double root). The parabola touches the x-axis tangentially at a single point. For example, x² + 4x + 4 = 0 has Δ = 16 - 16 = 0, yielding root x = -2.
3. Negative Discriminant (Δ < 0): The equation has no real solutions; instead, it has two complex conjugate roots in the format p ± qi, where i = √(-1). The parabola floats entirely above or below the x-axis without ever intersecting it. For x² + 2x + 5 = 0, Δ = 4 - 20 = -16, yielding complex roots -1 + 2i and -1 - 2i.
Parabolic Geometry: Beyond finding roots, analyzing the vertex coordinates (h, k) is essential. The x-coordinate of the vertex is h = -b / (2a), which also defines the vertical Axis of Symmetry (x = h) bisecting the parabola. The y-coordinate is k = c - b² / (4a). If the leading coefficient a is positive, the parabola opens upward and the vertex is a Minimum point. If a is negative, the parabola opens downward and the vertex is a Maximum point.
Linear Degeneracy: If a user enters a = 0, the second-degree term vanishes and the expression reduces to a linear equation: bx + c = 0. Rather than failing or returning NaN, our solver automatically catches this edge case, displays an informative notice, and solves the linear equation x = -c / b with full derivation steps.
Practical Tips
- ✓Remember that if coefficient a = 0, the equation ceases to be quadratic and collapses into a linear equation bx + c = 0, which our solver handles seamlessly.
- ✓Check the sign of the discriminant first: if negative, the roots contain imaginary numbers (i = √-1) and the parabola never crosses the x-axis.
- ✓The axis of symmetry is always the vertical line passing directly through the x-coordinate of the vertex: x = -b / (2a).
Common Mistakes to Avoid
- ⚠Forgetting parentheses when squaring negative b coefficients. (-5)² = +25, not -25.
- ⚠Dividing only the radical term by 2a instead of dividing the entire numerator (-b ± √Δ).
Related Guides & Calculation Deep Dives
All guidesQuadratic Solver FAQs
Common questions, calculation formulas, and usage rules.
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). It provides the exact solutions (roots) for any quadratic equation in standard form ax² + bx + c = 0.
If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has exactly one repeated real root. If Δ < 0, it has two complex conjugate roots involving imaginary numbers (i).