Quadratic Formula Calculator

Quadratic Equation Solver

Enter the coefficients a, b, and c. Fractional values (e.g., 3/4) are accepted. a cannot be zero.

ax² + bx + c = 0

a = b = c =

About This Quadratic Calculator

This Quadratic Formula Calculator solves any second-degree polynomial equation of the form ax² + bx + c = 0. Using the quadratic formula, it finds real and complex roots, calculates the discriminant, and shows step-by-step solutions. Perfect for students, teachers, and professionals working with algebra and parabolic functions.

How to Use

1. Enter Coefficient a — The coefficient of x² (cannot be zero). Supports fractions like 3/4.

2. Enter Coefficient b — The coefficient of x.

3. Enter Coefficient c — The constant term.

4. Click Solve Equation — Press the "Solve Equation" button to see the roots and step-by-step solution.

5. Review the Result — The calculator shows real or complex roots based on the discriminant value (Δ = b² - 4ac).

How It Works & Formula

The quadratic formula solves any equation of the form ax² + bx + c = 0 by finding where the parabola crosses the x-axis. The formula uses the discriminant to determine the nature of the roots.

x = [-b ± √(b² - 4ac)] / (2a)

Where:
• a = Coefficient of x² (a ≠ 0)
• b = Coefficient of x
• c = Constant term
• ± = Plus-minus symbol (two solutions)

Discriminant: Δ = b² - 4ac

Root types based on discriminant:
• Δ > 0 → Two distinct real roots
• Δ = 0 → One real root (double/repeated root)
• Δ < 0 → Two complex conjugate roots

Alternative Forms

x = [−b ± √Δ] / (2a)

When Δ > 0, the roots are real and the parabola crosses the x-axis at two points. When Δ = 0, the parabola touches the x-axis at one point (vertex). When Δ < 0, the parabola does not intersect the x-axis.

Real-Life Examples

Example 1: Solve x² + 5x + 6 = 0 (a=1, b=5, c=6)

Δ = b² - 4ac = 25 - 24 = 1 > 0
√Δ = 1
x = [-5 ± 1] / (2 × 1)
x₁ = (-5 + 1)/2 = -4/2 = -2
x₂ = (-5 - 1)/2 = -6/2 = -3

The equation factors as (x + 2)(x + 3) = 0, with roots at x = -2 and x = -3.

Example 2: Solve x² - 4x + 4 = 0 (a=1, b=-4, c=4)

Δ = (-4)² - 4 × 1 × 4 = 16 - 16 = 0
x = [4 ± 0] / (2 × 1) = 2

This perfect square trinomial factors as (x - 2)² = 0, with one repeated root at x = 2.

Example 3: Solve x² + x + 1 = 0 (a=1, b=1, c=1)

Δ = 1 - 4 = -3 < 0
√Δ = i√3
x = [-1 ± i√3] / 2 = -0.5 ± 0.866i

When the discriminant is negative, the quadratic has two complex conjugate roots.

Tip: Quadratic equations appear in physics (projectile motion), engineering (parabolic reflectors), economics (profit maximization), and many other fields.

Discriminant Reference Table

The discriminant (Δ = b² - 4ac) determines the nature of the roots.

Discriminant (Δ)	Root Type	Graph Behavior	Example
Δ > 0	Two distinct real roots	Parabola crosses x-axis at 2 points	x² - 3x + 2 = 0 → x = 1, 2
Δ = 0	One real root (double root)	Parabola touches x-axis at 1 point (vertex)	x² - 4x + 4 = 0 → x = 2
Δ < 0	Two complex conjugate roots	Parabola does not intersect x-axis	x² + x + 1 = 0 → x = -0.5 ± 0.866i

For Δ > 0, roots can be rational (perfect square) or irrational (non-perfect square). For Δ = 0, the quadratic is a perfect square trinomial.

Frequently Asked Questions

What is the quadratic formula? +

The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). It provides the solutions to any quadratic equation in the form ax² + bx + c = 0, where a ≠ 0. The formula has been known for thousands of years and is a fundamental tool in algebra.

What does the discriminant tell me? +

The discriminant (Δ = b² - 4ac) determines the nature of the roots. If Δ > 0, you get two real roots. If Δ = 0, you get one real root. If Δ < 0, you get two complex roots. The discriminant also tells you if the roots are rational (when Δ is a perfect square) or irrational.

Why can't 'a' be zero? +

If a = 0, the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula would require division by zero (2a = 0), making it undefined. Linear equations have only one solution and are solved differently.

What are complex roots? +

Complex roots occur when the discriminant is negative. They take the form p ± qi, where i = √(-1). These roots always come in conjugate pairs. Geometrically, complex roots mean the parabola never touches the x-axis.

Can I use fractions for coefficients? +

Yes! This calculator accepts fractions in the form "3/4", "-2/5", or "1/2". It converts them to decimals automatically and solves the equation accurately.

What is the vertex of a quadratic function? +

The vertex is the highest or lowest point of the parabola. Its x-coordinate is at x = -b/(2a). This is also the axis of symmetry for the parabola. For a positive a, the parabola opens upward and the vertex is a minimum; for negative a, it opens downward and the vertex is a maximum.

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