Quadratic Formula Calculator

Math

Solve any quadratic equation ax² + bx + c = 0 with real and complex roots, discriminant analysis, vertex coordinates, and step-by-step solutions.

Presets:

2 Real Roots

Root x₁ 3

Root x₂ 2

Discriminant (Δ) 1

Vertex (h, k) (2.50, -0.25)

Axis of Symmetry x = 2.50

Opens upward

Parabola Graph y = 1x² -5x + 6

Quadratic Formula Step-by-Step Derivation

1. Calculate the Discriminant (Δ)

Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

The discriminant determines the nature and quantity of the roots.

2. Apply the Quadratic Formula (Two Real Roots)

x = (-b ± √Δ) / 2a = (-(-5) ± √1) / (2 × 1) = (5 ± 1.0000) / 2

Δ > 0 yields two distinct real solutions: x₁ = 3, x₂ = 2.

How This Calculator Works

The quadratic formula is a universal algebraic formula used to find the solutions (roots or x-intercepts) of any second-degree polynomial equation in standard form: ax² + bx + c = 0 (where a ≠ 0). It was derived by completing the square on the general quadratic equation.

📐 The Quadratic Formula & Discriminant

x = (-b ± √(b² - 4ac)) / (2a) | Δ = b² - 4ac

a: Quadratic coefficient (determines parabola width and direction)

b: Linear coefficient (affects axis of symmetry)

c: Constant term (y-intercept of the parabola)

Δ: Discriminant: determines the number and nature of roots

Key Mathematical Rules & Properties

Two Distinct Real Roots (Δ > 0)

When the discriminant is positive, the square root yields a real positive number, giving two distinct real x-intercepts.

One Double Real Root (Δ = 0)

When the discriminant is zero, the parabola vertex touches the x-axis at exactly one point: x = -b / (2a).

Two Complex Conjugate Roots (Δ < 0)

When the discriminant is negative, the square root yields an imaginary number i = √(-1), producing complex roots u ± vi.

Parabola Vertex & Axis of Symmetry

The vertex represents the extreme minimum (a > 0) or maximum (a < 0) at x = -b / (2a), with axis of symmetry x = -b / (2a).

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Worked Example: Solving x² - 5x + 6 = 0

Find the real roots of the quadratic equation x² - 5x + 6 = 0.

Identify coefficients: a = 1, b = -5, c = 6
Calculate discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
Since Δ = 1 > 0, there are two distinct real roots
Apply quadratic formula: x = (-(-5) ± √1) / (2·1) = (5 ± 1) / 2
Root 1: x₁ = (5 + 1) / 2 = 6 / 2 = 3
Root 2: x₂ = (5 - 1) / 2 = 4 / 2 = 2

Final Answer: x₁ = 3, x₂ = 2

Frequently Asked Questions

What happens if a = 0 in a quadratic equation?

If a = 0, the equation reduces to bx + c = 0, which is a linear equation rather than a quadratic. The quadratic formula cannot be applied because it would cause division by zero.

What do complex roots mean geometrically?

Complex roots mean the parabola never crosses or touches the horizontal x-axis. It floats entirely above (when a > 0) or below (when a < 0) the x-axis.

How does factoring compare to using the quadratic formula?

Factoring is faster when roots are integers or simple fractions, but the quadratic formula always works for any quadratic equation, including those with irrational or complex roots.

How do you find the vertex of a parabola from its equation?

The x-coordinate of the vertex is h = -b / (2a). The y-coordinate is k = c - b² / (4a) or f(h).

Understanding your calculation

Method, example and limitations

How the calculation works

For ax² + bx + c = 0 with a ≠ 0, the roots are (−b ± √(b² − 4ac)) / (2a). The discriminant determines the nature of the roots.

Example to check

x² − 5x + 6 = 0 has discriminant 1 and roots 2 and 3. Substituting either root makes the expression zero.

Before using the result

If a is zero the equation is not quadratic. A negative discriminant has no real roots; complex output depends on the tool’s supported mode.

Calculation methodology · Report a result that needs review