Quadratic Equation Solver

Free quadratic equation solver. Enter a, b, c for ax² + bx + c = 0 and get roots using the quadratic formula with step-by-step working.

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Quadratic Solver: ax² + bx + c = 0

a (x² coefficient)

b (x coefficient)

c (constant)

Comprehensive Guide: Quadratic Equations, The Discriminant & Vertex Geometry

Master the universal quadratic formula, discriminant root classification, factoring vs. completing the square, parabola vertex coordinates, and physics trajectories.

1. Mathematical Definition of Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x whose standard algebraic form is written as:

  ax² + bx + c = 0   (where a, b, c ∈ ℝ and a ≠ 0)

The term ax² is the quadratic term, bx is the linear term, and c is the constant term. The condition a ≠ 0 is mandatory; if a = 0, the degree collapses to 1, degenerating into a simple linear equation.

2. The Universal Quadratic Formula & Its Algebraic Derivation

Every quadratic equation can be solved analytically using the quadratic formula:

  x = (−b ± √(b² − 4ac)) ÷ 2a

Step-by-Step Derivation (Completing the Square):

Start with standard form: ax² + bx + c = 0.
Subtract c from both sides and divide through by a: x² + (b/a)x = −c/a.
Complete the square by adding (b / 2a)² = b² / (4a²) to both sides:
x² + (b/a)x + b²/(4a²) = b²/(4a²) − c/a = (b² − 4ac) / (4a²).
Express the left side as a binomial square: (x + b/(2a))² = (b² − 4ac) / (4a²).
Extract square roots: x + b/(2a) = ±√(b² − 4ac) / 2a.
Isolate x: x = (−b ± √(b² − 4ac)) / 2a. (Q.E.D.)

3. The Discriminant (Δ) & Geometric Nature of Roots

The radicand Δ = b² − 4ac is defined as the discriminant. It predetermines the nature of the roots and the geometric relationship of the parabola y = ax² + bx + c to the x-axis:

Case 1 (Δ > 0): Two Distinct Real Roots. The parabola intersects the x-axis at two distinct real coordinates x₁ and x₂.
Case 2 (Δ = 0): Exactly One Repeated Real Root (Double Root). The vertex of the parabola touches the x-axis tangentially at x = −b / 2a.
Case 3 (Δ < 0): Two Complex Conjugate Roots. The parabola never crosses the x-axis. Solutions take the form u ± v·i, where i = √(−1).

4. Graduated Step-by-Step Worked Problems

Problem 1: Rational Real Roots (2x² − 7x + 3 = 0)

Step 1: Identify coefficients: a = 2, b = −7, c = 3.

Step 2: Calculate discriminant:
Δ = (−7)² − 4(2)(3) = 49 − 24 = 25. (Since Δ = 25 > 0, two real roots exist, and since 25 is a perfect square, roots will be rational).

Step 3: Apply formula:
x = (−(−7) ± √25) / (2 × 2) = (7 ± 5) / 4.

Step 4: Compute individual roots:
• x₁ = (7 + 5) / 4 = 12 / 4 = 3.0
• x₂ = (7 − 5) / 4 = 2 / 4 = 0.5

Step 5: Parabola Vertex Coordinates:
• x_v = −b / 2a = 7 / 4 = 1.75
• y_v = −Δ / 4a = −25 / 8 = −3.125. Vertex at (1.75, −3.125).

Problem 2: Complex Conjugate Roots (x² − 4x + 13 = 0)

• Discriminant: Δ = (−4)² − 4(1)(13) = 16 − 52 = −36 < 0.
• √Δ = √(−36) = 6i.
• Roots: x = (4 ± 6i) / 2 = 2 ± 3i.
Result: Roots are x = 2 + 3i and x = 2 − 3i.

5. Real-World Applications in Ballistics, Engineering & Economics

Projectile Motion & Trajectory Physics: Under gravitational acceleration g, height follows h(t) = −(1/2)gt² + v₀t + h₀. Setting h(t) = 0 yields the time of impact.
Parabolic Solar Reflectors & Satellite Dishes: Any ray parallel to the axis of symmetry reflects directly into the parabola's geometric focus point (0, 1/(4a)).
Economic Revenue Optimization: Revenue functions are quadratic R(p) = −ap² + bp, where the vertex represents the profit-maximizing price.

6. Frequently Asked Questions (FAQ)

When should I factor instead of using the quadratic formula?

If the discriminant b² − 4ac is a perfect square (0, 1, 4, 9, 16, 25...), the equation factors cleanly over rational numbers. However, the quadratic formula works universally on every quadratic, including irrational and complex cases.

What are Vieta's formulas for quadratic equations?

Vieta's formulas establish that for roots r₁ and r₂: their sum r₁ + r₂ = −b / a, and their product r₁ · r₂ = c / a. This allows rapid mental verification of computed roots.

How do I know if a parabola opens upward or downward?

The sign of the leading coefficient a dictates the orientation. If a > 0, the parabola opens upward (U-shaped) and the vertex is an absolute minimum. If a < 0, it opens downward and the vertex is a maximum.