Math Equation Solver

Free Online Math Equation Solver

This math equation solver handles the three most common algebra problems with full step-by-step working: linear equations (ax + b = c), quadratic equations (ax² + bx + c = 0), and systems of two linear equations. Type in your coefficients — the solver gives you the answer immediately, plus the reasoning, so it's actually useful for homework and learning, not just for the final number.

How to Use the Math Solver

Pick the equation type from the tabs at the top, type your coefficients into the boxes, and the answer appears as you type. For linear equations, enter the coefficient of x (a), the constant added on the left (b), and the right-hand side (c). For quadratic equations, enter a, b, c from ax² + bx + c = 0. For systems of two equations, enter all six coefficients for the two lines. Each result includes the formula used and a step-by-step breakdown.

Linear Equations: ax + b = c

The simplest equation type. The solver isolates x in one step:

1. Subtract b from both sides: ax = c − b
2. Divide both sides by a: x = (c − b) ÷ a

Special cases: If a = 0 and b = c, every value of x works (infinite solutions). If a = 0 and b ≠ c, no value of x works (no solution). The solver detects and explains both cases.

Quadratic Equations: ax² + bx + c = 0

Solved with the quadratic formula:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant D = b² − 4ac tells you immediately what kind of solutions to expect:

• D > 0 → two distinct real solutions (parabola crosses the x-axis at two points)
• D = 0 → one repeated real solution (parabola just touches the x-axis)
• D < 0 → two complex conjugate solutions (parabola never touches the x-axis); the solver reports them as real ± imag·i

The solver also computes the vertex (the parabola's turning point) at x = −b/(2a).

System of 2 Linear Equations

Given:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

The solver uses Cramer's rule — a clean, foolproof algorithm using 2×2 determinants:

• D = a₁b₂ − a₂b₁ (the main determinant)
• Dx = c₁b₂ − c₂b₁ (replace x's column with c)
• Dy = a₁c₂ − a₂c₁ (replace y's column with c)
• x = Dx / D, y = Dy / D

Special cases: If D = 0 and Dx = Dy = 0, the two equations are the same line (infinite solutions). If D = 0 but Dx or Dy is non-zero, the lines are parallel (no solution).

Where the Math Solver Is Useful

• Algebra homework — check your work against the solver's step-by-step
• Physics problems — solving for velocity, time, force using kinematic equations
• Finance — finding break-even points (quadratic when there's a fixed + variable cost structure)
• Engineering — finding intersection points of two beams or surfaces (system of equations)
• Geometry — quadratic equations for finding lengths from area constraints
• Test prep — SAT, ACT, GRE all heavily test these three equation types

Frequently Asked Questions

How accurate is the solver?

It uses standard floating-point arithmetic — accurate to about 15 significant digits, which is more than enough for any homework, textbook problem, or real-world calculation. Tiny rounding artefacts near zero (like 1e-16 instead of 0) are auto-cleaned in the display.

Can it handle equations with x³ or higher powers?

This solver covers linear (x¹) and quadratic (x²) only. Cubic and quartic equations need different methods (Cardano's formula / Ferrari's formula) and are out of scope for this tool.

What if my equation has fractions or decimals?

Just enter the decimal value of each coefficient. The solver handles any finite real number — fractions, decimals, negatives, all fine. For example, "½x + ⅓ = ¾" becomes a=0.5, b=0.3333..., c=0.75.

What does the discriminant tell me?

The discriminant (D = b² − 4ac) tells you the nature of a quadratic's solutions before you compute them: positive = two real solutions, zero = one repeated solution, negative = no real solutions (complex only). It's the single most useful thing to look at when sketching a parabola.

Why doesn't it just take a typed equation like "2x + 3 = 7"?

String parsing is fragile — every variation ("2 x + 3 = 7", "2*x+3=7", "2(x) + 3 = 7", "x*2 + 3 = 7") would need handling. Explicit coefficient inputs are unambiguous, work on phones, and never misinterpret what you meant.