Overview & Concepts
An equation is a mathematical statement that two expressions are equal, and 'solving' it means finding the value (or values) of the unknown variable that make the statement true. Our calculator equation solver handles the most common equation types you'll meet in school and everyday work: linear equations (where the variable appears only to the first power), quadratic equations (where it appears squared), and simple rational or radical equations.
As a general-purpose calculator math solver, it goes beyond just a final numeric answer — it shows the full working so you can see exactly how the variable was isolated, or how the quadratic formula was applied. This makes it useful for checking homework, studying for exams, or verifying calculations for engineering and finance work.
If your equation involves a matrix or a system of several equations at once, our Matrix & Vector Calculator can handle row reduction and systems in one pass. And if you need to go beyond solving for x into rates of change or areas under curves, check out the Calculus Solver.
Step-by-Step Mathematical Method
Linear Equations: An equation like ax + b = c is solved by isolating x: subtract b from both sides, then divide both sides by a (provided a ≠ 0), giving x = (c − b) / a.
Quadratic Equations: For a standard-form equation ax2 + bx + c = 0 (with a ≠ 0), the quadratic formula gives x = (−b ± √(b2 − 4ac)) / (2a). This is the basis of any quadratic equation solver with working, since it always produces both possible solutions in one formula.
The Discriminant: The expression under the square root, b2 − 4ac, is called the discriminant. If it's positive, the equation has two distinct real solutions; if it equals zero, there is exactly one repeated real solution; if it's negative, the two solutions are complex (involving the imaginary unit i).
Factoring: Some quadratics can be solved faster by factoring into two binomials, e.g. x2 + 5x + 6 = (x + 2)(x + 3), then setting each factor equal to zero. This only works cleanly when the roots are rational.
Worked Examples with Solutions
Example 1: Linear Equation
Linear Equation Walkthrough
Solve 3x + 7 = 22, copy each step directly:
Example 2: Quadratic Equation via the Quadratic Formula
Quadratic Formula Walkthrough
Solve 2x^2 - 4x - 6 = 0, copy each step directly:
Example 3: Solving by Factoring
Factoring Walkthrough
Solve x^2 - x - 12 = 0, copy each step directly:
Reference Values & Comparison Table
| Equation Type | Formula Used | Result (from examples above) |
|---|---|---|
| Linear: 3x + 7 = 22 | x = (c - b) / a | x = 5 |
| Quadratic: 2x^2 - 4x - 6 = 0 | x = (-b ± √(b²-4ac)) / 2a | x = 3 or x = -1 |
| Factorable Quadratic: x^2 - x - 12 = 0 | Factor into (x - p)(x - q) | x = 4 or x = -3 |
Common Mistakes & Tips
Other frequent errors include forgetting the ± symbol in the quadratic formula (which loses one of the two valid solutions), dividing by a variable expression without first checking whether it could equal zero, and forgetting to distribute a negative sign across parentheses when simplifying both sides of an equation.
Always check your final answer by substituting it back into the original equation — this is the fastest way to catch an arithmetic slip. If you're working with a calculator device rather than the browser tool, our guide on how to use the equation solver on a calculator walks through the keystrokes for common models.