Equation Solver

Solve quadratic equations and systems of linear equations with step-by-step solutions.

ax² + bx + c = 0

a₁x + b₁y = c₁
a₂x + b₂y = c₂

x + y =

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

x + y + z =

How It Works

Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. Solutions are found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

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

Δ > 0 — Two distinct real roots
Δ = 0 — One repeated real root
Δ < 0 — Two complex conjugate roots

Systems of Linear Equations

A system of linear equations is solved using Cramer's Rule, which expresses each variable as a ratio of determinants. For a 2×2 system:

x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁)
y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁)

The system has a unique solution when the determinant of the coefficient matrix is non-zero. If the determinant is zero, the system is either inconsistent (no solution) or dependent (infinitely many solutions).

Vertex Form

Every quadratic equation can be rewritten in vertex form: a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex is at h = -b/(2a) and k = f(h).

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