Quadratic Equation Solver

The solver follows the standard quadratic equation solution process, enhanced with exact radical simplification, fraction reduction, and clear formatting.

1. Equation Parsing

The input is split around =0 and scanned for the x², x, and constant terms using regular expressions. It supports:

• Optional + and - signs
• Decimal coefficients
• Implicit coefficients (for example, x^2 is treated as 1x^2)

If the equation cannot be parsed, the solver returns a clear error message.

2. Coefficient Extraction

The coefficients a, b, and c are extracted as follows:

• a → coefficient of the x² term
• b → coefficient of the x term (excluding x²)
• c → constant term

3. Discriminant Calculation

The discriminant (Δ) is calculated using:

Δ = b² − 4ac

The value of the discriminant determines the type of roots:

• Δ > 0 → Two distinct real roots
• Δ = 0 → One repeated real root (double root)
• Δ < 0 → Two complex conjugate roots

4. Root Calculation

The roots are calculated using the quadratic formula:

x = (−b ± √Δ) / (2a)

If the discriminant is negative, the roots become:

x = (−b ± i√|Δ|) / (2a)

5. Exact Form Simplification

When all coefficients are integers, the solver simplifies radicals by factoring out perfect squares. For example:

√12 = 2√3

The resulting fraction is then reduced to lowest terms using the greatest common divisor (GCD), producing simplified expressions such as:

(1 ± 2√3) / 3

For decimal coefficients, exact radical forms are often not meaningful, so the solver displays decimal approximations instead.

6. Vertex & Axis of Symmetry

For the parabola:

y = ax² + bx + c

the vertex (h, k) is calculated as:

h = −b / (2a)

k = c − b² / (4a)

The axis of symmetry is the vertical line:

x = h

7. Factored Form

When the discriminant is a perfect square and the coefficients are integers, the equation is written in factored form:

a(x − r₁)(x − r₂)

where r₁ and r₂ are the roots.

8. Edge Cases

The solver also handles special cases automatically:

• a = 0 and b ≠ 0 → Solves the equation as a linear equation:

x = −c / b

• a = 0 and b = 0 → Treated as a constant equation:
• c = 0 → Infinitely many solutions
• c ≠ 0 → No solution