Algebra Calculator
This Algebra Calculator helps you solve for unknowns and make sense of common algebra tasks without guesswork—linear equations, quadratics, 2×2 systems, and quadratic factoring. Use it when you want clean steps, substituted values, and a final answer you can verify. If you're exploring more tools, you can browse the full calculator library or jump straight into math-focused calculators.
Solve, Factor, and Evaluate
Pick a mode, enter values, then press Calculate. Use arrow keys on the mode selector for quick switching.
Solve
ax + b = c. This coefficient method is reliable and avoids ambiguous equation parsing.
Example: in
2x + 5 = 17, a = 2Example: in
2x + 5 = 17, b = 5Example: in
2x + 5 = 17, c = 17
Solve
ax² + bx + c = 0 using the discriminant and the quadratic formula.
If a = 0, the equation becomes linear.
Careful with the sign of b.
Constant term in
ax² + bx + c.
Solve a system using Cramer’s Rule. Enter coefficients for:
a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
Factor
ax² + bx + c using GCF and the AC method when possible.
Evaluate a safe arithmetic expression using only digits, parentheses, + − × ÷, and decimals.
Allowed characters: 0–9, (), +, -, *, /, . and spaces.
Calculations run locally in your browser. Nothing is sent anywhere.
How it works
Each mode uses standard algebra formulas. For a linear equation ax + b = c, we isolate the variable:
x = (c − b) / a. Quadratics use the quadratic formula
x = (−b ± √(b² − 4ac)) / 2a, where the discriminant D = b² − 4ac tells us how many real roots exist.
Systems use the determinant Δ = a₁b₂ − a₂b₁. If Δ ≠ 0, there is one unique solution. Factoring looks for common factors first, then applies the AC method to split the middle term.
Looking for more equation tools? Explore the complete Math Calculators hub.
Use cases
- Budget planning where costs and revenue form simple linear equations.
- Physics problems like projectile height modeled by quadratic functions.
- Mixture or rate problems that naturally create two equations with two variables.
- Break-even analysis in business models.
- Geometry relationships that translate into algebraic equations.
Worked examples
Linear: 2x + 5 = 17 → 2x = 12 → x = 6
Quadratic: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2, 3
System: 2x + y = 7, x − y = 1 → x = 8/3, y = 5/3
Common mistakes
- Forgetting to move constants before dividing.
- Sign errors with the coefficient b.
- Treating a = 0 as quadratic.
- Mixing equation coefficients in systems.
- Rounding too early.
Quick tips
- Keep fractions exact until the end.
- Check the discriminant first.
- Use parentheses when substituting.
- Normalize equations before solving.
- Verify by plugging answers back in.
FAQ
What is a discriminant and why it matters?⌄
The discriminant is the value b² − 4ac in the quadratic formula. It tells you how many real solutions exist before you even compute square roots. A positive value means two distinct real roots, zero means one repeated root, and a negative value produces complex conjugate roots. Checking it first helps you understand the nature of the solutions and choose the correct solving approach.
Why do I get “no solution” for a linear equation?⌄
If the coefficient of x becomes zero while constants remain different, the equation reduces to something impossible like 5 = 3. That means no value of x can satisfy the statement. This happens when both sides are parallel lines or inconsistent constraints. The calculator flags this clearly instead of dividing by zero.
What does determinant zero mean in a system?⌄
A determinant of zero indicates the equations are dependent or parallel. Dependent equations represent the same line and have infinitely many solutions, while parallel ones never meet and have no solution. In both cases, Cramer’s Rule cannot produce a unique answer, so the calculator explains the situation instead of giving misleading numbers.
How does factoring relate to roots?⌄
Factoring rewrites a quadratic into products like (x − r₁)(x − r₂). Setting each factor equal to zero reveals the roots directly. This approach is often faster than the quadratic formula when integer factors exist. The calculator attempts factoring first for clarity and falls back to roots if necessary.
Can the calculator handle fractions or decimals?⌄
Yes. You can enter decimals or numbers with commas, and the parser converts them safely. Internally calculations use floating-point math, and results are rounded to six decimal places for readability while still keeping precision high enough for most school and practical problems.
Why are my answers slightly different from a textbook?⌄
Small differences often come from rounding or using decimals instead of exact fractions. Textbooks may keep symbolic or fractional results longer. The calculator rounds only at the end to reduce error, but tiny variations can still appear depending on precision limits.
What are complex roots and when do they appear?⌄
Complex roots occur when the discriminant is negative. Since you cannot take the square root of a negative number within real numbers, the solutions include the imaginary unit i. They appear as a ± bi and represent points off the real number line but still satisfy the equation algebraically.
How do I format my equation inputs or coefficients?⌄
Enter clean numbers such as 2, −3.5, or 1,200. Avoid letters or symbols. For equations, use the coefficient boxes provided for accuracy. This prevents parsing errors and ensures the calculator applies the correct formulas and steps for each solving method.
More Helpful Calculators
Rounding: Results shown to 6 decimals unless noted.
Privacy: No inputs are stored or transmitted.
Last updated: January 26, 2026