Algebra Calculator
Solve linear and quadratic equations step by step
Algebra Calculator
Solve quadratic equations ax² + bx + c = 0
Find roots of ax² + bx + c = 0
x = (-b +/- sqrt(b² - 4ac)) / 2aWhat is an Algebra Calculator?
An Algebra Calculator is a math tool that helps you work with algebraic expressions and equations quickly and accurately. Algebra is the branch of mathematics that uses variables (like x, y, or z) to represent unknown values and uses rules to simplify expressions, solve equations, factor polynomials, and expand or rearrange terms.
Instead of doing long manual steps—like combining like terms, distributing parentheses, or solving for a variable—an algebra calculator can perform these operations instantly. This is especially helpful for checking homework, verifying steps in a math problem, or exploring 'what-if' scenarios by changing values.
Algebra calculators are commonly used in middle school and high school math (pre-algebra, algebra 1/2), as well as in college courses like calculus, physics, chemistry, economics, and engineering—anywhere equations and formulas need to be simplified or solved.
How to Use This Algebra Calculator
- Enter your expression or equation -- Example expressions: 3x + 2x - 7 or 2(x + 4) - 3x. Example equations: 2x + 5 = 17.
- Choose the operation (if applicable) -- such as Simplify, Solve, Factor, Expand, or Evaluate.
- Select the variable (if applicable) -- for example, solve for x.
- Click 'Calculate' -- the calculator will produce the simplified form or solution.
- Review the result -- some calculators also show steps; if shown, use them to learn the process.
Tips:
- Use parentheses to clearly group terms: 2(x + 3)
- Use ^ for exponents if supported: x^2
- If you get an unexpected result, double-check signs and parentheses (most mistakes come from missing parentheses or negative signs)
Algebra Formulas
Combining Like Terms
Like terms have the same variable part (same variables raised to the same powers):
- 3x + 2x = 5x
- 7a² − 4a² = 3a²
Distributive Property
Rule: a(b + c) = ab + ac
Example: 2(x + 5) = 2x + 10
Solving a Linear Equation
General form: ax + b = c
Solve for x: x = (c − b) / a
Isolate x by subtracting b, then dividing by a
Factoring a Quadratic
Form: x² + bx + c
Find two numbers that multiply to c and add to b:
x² + bx + c = (x + m)(x + n)
Quadratic Formula (Solving ax² + bx + c = 0)
x = (−b ± √(b² − 4ac)) / 2a
The expression b² − 4ac is the discriminant, which determines the number and type of solutions:
- Discriminant > 0: two distinct real roots
- Discriminant = 0: one repeated real root
- Discriminant < 0: two complex (imaginary) roots
Example Calculations
Example 1: Simplify an expression
Expression: 3x + 2x − 7
Step: Combine like terms: 3x + 2x = 5x
Answer: 5x − 7
Example 2: Expand using distribution
Expression: 2(x + 4) − 3x
Step 1: Distribute: 2(x + 4) = 2x + 8
Step 2: Subtract 3x: (2x + 8) − 3x = −x + 8
Answer: 8 − x
Example 3: Solve a linear equation
Equation: 2x + 5 = 17
Step 1: Subtract 5 from both sides: 2x = 12
Step 2: Divide by 2: x = 6
Answer: x = 6
Example 4: Factor a quadratic
Expression: x² + 5x + 6
Step: Find two numbers that multiply to 6 and add to 5 → 2 and 3
Answer: (x + 2)(x + 3)
Frequently Asked Questions
What is a variable in algebra?
A variable is a symbol (like x or y) that represents an unknown or changeable value. For example, in 2x + 3, the value of x can vary.
What does it mean to 'simplify' an expression?
Simplifying means rewriting an expression in a cleaner form by combining like terms, reducing fractions, and removing unnecessary parentheses—without changing its value.
What’s the difference between an expression and an equation?
An expression does not have an equals sign (example: 3x + 2). An equation includes an equals sign and states two things are equal (example: 3x + 2 = 11).
Why do I need parentheses?
Parentheses show grouping and control the order of operations. For example, 2(x + 3) is different from 2x + 3.
Can an algebra calculator solve any equation?
Many can solve common types (linear, some quadratics, basic systems), but very complex equations may have restrictions depending on the tool. If your equation doesn’t solve, try simplifying it first or confirm the calculator supports that equation type.
Want to add this algebra calculator to your website? Get a custom embed code that matches your site's design and keeps visitors engaged.
What Is Algebra?
Algebra is the branch of mathematics that uses symbols — usually letters like x and y — to represent unknown quantities and relationships between them. It forms the backbone of virtually every quantitative field: science relies on it for formulas, engineering uses it to model systems, finance applies it to calculate returns and risk, and computer programming depends on it for algorithms and logic. Understanding algebra means understanding how to work with unknowns and express rules in a general, reusable way.
Solving an equation means finding the specific value of x (or any variable) that makes both sides equal. Linear equations produce exactly one solution because the variable appears only to the first power. Quadratic equations — where the variable is squared — can have zero, one, or two real solutions depending on whether the parabola they represent crosses, touches, or misses the x-axis entirely. This calculator handles both types and shows you exactly how to get there.
How to Use the Algebra Calculator
- Select the equation type — Linear (ax + b = 0) or Quadratic (ax² + bx + c = 0).
- Enter the coefficients a, b, and c in their respective fields. For linear equations, only a and b are needed.
- Click the Solve button to compute the solution(s).
- Read the result — the calculator shows each root and, for quadratics, the value of the discriminant.
Formulas Used
Linear: ax + b = 0 → x = -b / a
Quadratic: ax² + bx + c = 0
x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Δ > 0 → two distinct real roots
Δ = 0 → one real root (double root)
Δ < 0 → no real roots (complex roots)The quadratic formula is universal — it works for any quadratic equation regardless of whether it factors neatly. When the discriminant Δ is negative, the equation has no real solutions; the roots are complex numbers involving the imaginary unit i.
Worked Examples
Example 1 — Linear: 2x + 6 = 0
Rearrange to standard form: 2x + 6 = 0, so a = 2, b = 6. Applying x = -b / a gives x = -6 / 2 = -3. Check: 2(-3) + 6 = -6 + 6 = 0. Correct.
Example 2 — Quadratic: x² - 5x + 6 = 0
Coefficients: a = 1, b = -5, c = 6. Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. Since Δ > 0, there are two real roots: x = (5 ± 1) / 2, giving x = 3 and x = 2. Check: (3-2)(3-3) — both satisfy the equation.
Example 3 — No Real Solution: x² + 4 = 0
Coefficients: a = 1, b = 0, c = 4. Discriminant: Δ = 0² - 4(1)(4) = -16. Since Δ < 0, there are no real solutions. The roots are the complex numbers x = ±2i.