Quadratic Equation Solver

Quadratic Equation Solver is evaluated from a, b and c. The calculation reports Discriminant, Root x₁ and Root x₂.

Results

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About the Quadratic Equation Solver

Quadratic Equation Solver is treated here as a quantitative relation between a, b and c and Discriminant, Root x₁, Root x₂ and Nature of Roots.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
x = ( - b ± sqrt(b^2 - 4ac)) / 2a
Vertex: ( - b/2a, c - b^2/4a)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

x = ( - b ± sqrt(b^2 - 4ac)) / 2a
Vertex: ( - b/2a, c - b^2/4a)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Two Real Roots — Projectile Height

Inputs

a_coef: -16 b_coef: 64 c_coef: 0
Discriminant: 4,096. Root x₁: 0. Root x₂: 4. Nature of Roots: Two distinct real roots. Vertex x: 2. Vertex y: 64

With a = -16, b = 64 and c = 0 as the stated inputs, the result is Discriminant = 4,096, Root x₁ = 0 and Root x₂ = 4. Each value corresponds to the declared output fields.

Example 2: Factoring — Integer Roots

Inputs

a_coef: 1 b_coef: -5 c_coef: 6
Discriminant: 1. Root x₁: 3. Root x₂: 2. Nature of Roots: Two distinct real roots. Vertex x: 2.5. Vertex y: -0.25

With a = 1, b = -5 and c = 6 as the stated inputs, the result is Discriminant = 1, Root x₁ = 3 and Root x₂ = 2. Each value corresponds to the declared output fields.

Example 3: Repeated Root — Perfect Square

Inputs

a_coef: 1 b_coef: -6 c_coef: 9
Discriminant: 0. Root x₁: 3. Root x₂: 3. Nature of Roots: One repeated real root (discriminant = 0). Vertex x: 3. Vertex y: 0

With a = 1, b = -6 and c = 9 as the stated inputs, the result is Discriminant = 0, Root x₁ = 3 and Root x₂ = 3. Each value corresponds to the declared output fields.

Example 4: No Real Roots — Complex Solution

Inputs

a_coef: 1 b_coef: 2 c_coef: 5
Discriminant: -16. Root x₁: -1. Root x₂: -1. Nature of Roots: Two complex conjugate roots (no real solution). Vertex x: -1. Vertex y: 4

With a = 1, b = 2 and c = 5 as the stated inputs, the result is Discriminant = -16, Root x₁ = -1 and Root x₂ = -1. Each value corresponds to the declared output fields.

Common Use Cases

  • Solve ax² + bx + c = 0 for x
  • Find parabola vertex and axis of symmetry
  • Determine if equation has real or complex roots
  • Factor a quadratic expression