Algebra Calculator

Solve linear equations, quadratic equations, simplify expressions, and systems of equations with detailed step-by-step solutions and explanations.

What Is Algebra?

Algebra is the branch of mathematics that uses symbols, typically letters, to represent unknown values and express relationships between quantities. An equation like 2x + 3 = 7 states that some number x, when doubled and increased by 3, equals 7. Solving the equation means finding the value of x that makes this statement true.

Algebraic thinking is foundational to science, engineering, economics, and computer science. Every time you set up a formula and solve for an unknown quantity you are doing algebra. This calculator automates the mechanical steps so you can focus on understanding the concepts and setting up problems correctly.

How the Math Works

Linear equations (ax + b = c) are solved by isolating x. Subtract b from both sides to get ax = c - b, then divide by a to find x = (c - b) / a. The process is straightforward unless a = 0, which makes the equation either always true (infinite solutions) or never true (no solution).

Quadratic equations (ax + bx + c = 0) are solved using the quadratic formula. The key is the discriminant, b squared minus 4ac. This value determines the nature of the solutions before you even compute them.

Expression simplification combines like terms by grouping all terms with the same variable and summing their coefficients. Constants are grouped separately. The result is a cleaner, equivalent expression.

Systems of two linear equations can be solved by Cramer's rule, substitution, or elimination. This calculator uses Cramer's rule, which computes the answer using determinants. The main determinant must be nonzero for a unique solution to exist.

How to Use This Calculator

Select the calculation type from the dropdown menu. The input fields change to match your selection.
Enter coefficients for your equation. For linear equations, fill in a, b, and c. For quadratic equations, provide the three coefficients. For systems, enter all six values across the two equations.
For simplification, type the algebraic expression directly into the text field using standard notation with x as the variable.
Read the result which includes the answer, the problem type, and a complete step-by-step solution showing every algebraic manipulation.
Check the discriminant for quadratic equations to understand the nature of the roots before examining the numeric values.

Worked Examples

Example 1: Solving a Linear Equation

Solve 3x + 7 = 22. Enter a = 3, b = 7, c = 22. The calculator subtracts 7 from both sides to get 3x = 15, then divides by 3 to get x = 5.

Example 2: Quadratic with Two Solutions

Solve x squared minus 5x plus 6 = 0. Enter a = 1, b = -5, c = 6. The discriminant is 25 - 24 = 1 (positive), so there are two solutions: x = 3 and x = 2.

Example 3: Simplifying an Expression

Simplify 4x - 2x + 8 - 3 + x. The calculator combines x-terms: 4 - 2 + 1 = 3x. Constants: 8 - 3 = 5. Result: 3x + 5.

Example 4: System of Two Equations

Solve 2x + 3y = 7 and x - y = 1. The determinant is 2(-1) - 1(3) = -5. Using Cramer's rule, x = 2 and y = 1. Verification: 2(2) + 3(1) = 7 and 2 - 1 = 1.

Common Use Cases

Homework and exams: Verify solutions to textbook algebra problems and understand the steps involved in solving each type.
Science and engineering: Solve for unknown variables in formulas such as velocity, force, or electrical resistance.
Finance: Determine break-even points, calculate required investment returns, and solve for unknown payment amounts in simple interest equations.
Programming: Debug mathematical logic by confirming expected outputs of algebraic formulas before coding them.
Test preparation: Practice solving different equation types and build confidence with step-by-step feedback.

Tips and Common Mistakes

Watch the signs of coefficients. When entering a quadratic like x squared minus 5x plus 6 = 0, the b coefficient is -5, not 5. A sign error changes the solutions entirely.

Make sure a is not zero in a quadratic. If a = 0 the equation is linear, not quadratic. The calculator handles this gracefully by solving the resulting linear equation instead.

Simplification only handles single-variable expressions. The current simplifier works with terms containing x and numeric constants. It does not handle multiple variables, exponents, or parenthesized groups.

Verify system solutions by substitution. The calculator shows a verification step, but it is good practice to mentally check that your x and y values satisfy both original equations.

Understand when no solution exists. A system with a zero determinant means the two equations represent parallel lines that never intersect, or the same line with infinitely many solutions.

Frequently Asked Questions

What types of equations can this calculator solve?

This calculator solves four types of algebra problems: linear equations in the form ax plus b equals c, quadratic equations in the form ax squared plus bx plus c equals zero, expression simplification by combining like terms, and systems of two linear equations with two unknowns using Cramer's rule.

What is the quadratic formula?

The quadratic formula is x equals negative b plus or minus the square root of b squared minus 4ac all divided by 2a. It solves any quadratic equation of the form ax squared plus bx plus c equals zero. The expression under the square root is called the discriminant and determines whether the equation has two, one, or zero real solutions.

What does the discriminant tell me?

A positive discriminant means the quadratic equation has two distinct real solutions. A discriminant of zero means there is exactly one repeated real solution. A negative discriminant means there are no real solutions and the roots are complex numbers. The calculator clearly labels which case applies to your equation.

How does the expression simplifier work?

The simplifier identifies terms containing x and constant terms separately, adds up the coefficients of all x-terms and adds up all constant values, then writes the combined result. For example 3x plus 2x minus 4 plus 7 simplifies to 5x plus 3 because the x-coefficients sum to 5 and the constants sum to 3.

What is Cramer's rule for solving a system of equations?

Cramer's rule uses determinants to solve systems of linear equations. For two equations with two unknowns it computes the main determinant and two auxiliary determinants by replacing columns with the constants. Each variable equals its auxiliary determinant divided by the main determinant, providing an exact algebraic solution.

Can the calculator handle equations with decimal coefficients?

Yes. All input fields accept decimal values such as 1.5 or negative 0.75. The calculator performs standard floating-point arithmetic on these values and displays results rounded to six decimal places. This makes it suitable for applied problems where coefficients come from real-world measurements.

What happens when a linear equation has no solution?

When the coefficient of x is zero and the constant terms do not balance, the equation is a contradiction with no solution. For example 0x plus 5 equals 3 is impossible. The calculator detects this case and displays a message explaining that no solution exists along with the reasoning.

Can I solve higher-degree polynomial equations?

This calculator supports linear and quadratic equations only. Cubic and higher-degree polynomials require more advanced methods such as numerical root-finding algorithms or specialized computer algebra systems. For basic algebra coursework covering first and second degree equations this tool covers the standard curriculum.