Algebra Calculator
Solve linear and quadratic equations with step-by-step solutions
Solution Steps
Enter values to see solution steps
How Does the Algebra Calculator Solve Equations?
The algebra calculator solves equations by applying algebraic operations to isolate the variable. For linear equations in the form ax + b = c, it subtracts b from both sides, then divides by a to find x. The calculator shows each step, helping students understand the solution process while verifying their own work.
Quadratic equations (ax² + bx + c = 0) use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The calculator computes the discriminant (b² - 4ac) first to determine whether solutions are real, then calculates both roots when they exist. Complex roots (when the discriminant is negative) indicate no real solutions.
What Types of Equations Can This Calculator Solve?
This calculator handles two fundamental equation types. Linear equations have one variable raised to the first power - these always have exactly one solution (unless a=0). Quadratic equations contain a squared variable and can have zero, one, or two real solutions depending on the discriminant value.
Linear equations appear in everyday situations like calculating costs, distances, or times. Quadratic equations model parabolic motion, area problems, and optimization scenarios. Mastering both types builds essential foundations for higher mathematics. For basic math operations, see our percentage calculator.
Discriminant and Quadratic Solutions
| Discriminant (b²-4ac) | Number of Solutions | Type |
|---|---|---|
| Positive (> 0) | Two | Two distinct real roots |
| Zero (= 0) | One | One repeated real root |
| Negative (< 0) | None (real) | Two complex conjugate roots |
When Should You Use the Quadratic Formula?
The quadratic formula works for any quadratic equation regardless of whether it factors nicely. While factoring can be faster for simple equations, the formula provides a reliable method when factoring isn't obvious. It's especially useful when coefficients are decimals, fractions, or large numbers that make mental factoring impractical.
When solving word problems, the quadratic formula handles the resulting equations regardless of complexity. Physical applications like projectile motion, area optimization, and revenue maximization frequently produce non-factorable quadratics where the formula becomes essential.
Common Algebra Equation Examples
Linear example: 3x + 7 = 22 solves to x = 5 by subtracting 7 (gives 3x = 15) then dividing by 3. Quadratic example: x² - 4x + 3 = 0 factors to (x-1)(x-3) = 0, giving solutions x = 1 and x = 3. The calculator handles both approaches, showing detailed steps for educational value.
Frequently Asked Questions
What is an algebra calculator?
An algebra calculator solves equations by finding the value of unknown variables. Enter the coefficients and constants from your equation, and the calculator applies algebraic rules to determine the solution, showing step-by-step work for linear and quadratic equations.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, used to solve equations in the form ax² + bx + c = 0. The ± symbol means calculating two solutions: one using addition and one using subtraction of the square root term.
Why does my quadratic equation have no real solutions?
When the discriminant (b² - 4ac) is negative, the equation has no real solutions because you cannot take the square root of a negative number in real numbers. The solutions exist as complex numbers involving the imaginary unit i, but these aren't shown on a standard number line.
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Last Updated: January 2026 | Reviewed for accuracy