What Is an Algebra Solver?
An algebra solver helps you find the unknown value (usually called x) that makes an equation true. Whether you're solving a simple equation like 2x + 5 = 13 or a complex quadratic like x² - 4x + 3 = 0, the process follows the same basic principle: isolate the variable on one side.
Types of Equations You Can Solve
| Type | Example | Method |
|---|---|---|
| One-step | x + 5 = 12 | Subtract 5 |
| Two-step | 3x - 7 = 14 | Add 7, divide by 3 |
| Multi-step | 4(x + 2) = 3x + 11 | Distribute, combine, isolate |
| Quadratic | x² - 5x + 6 = 0 | Factor, quadratic formula |
| System of equations | x + y = 10, 2x - y = 5 | Substitution, elimination |
Solving Linear Equations Step by Step
Example: Solve 4(x + 2) = 3x + 11
- Distribute: 4x + 8 = 3x + 11
- Move variables to one side: 4x - 3x + 8 = 11 → x + 8 = 11
- Move constants to the other side: x = 11 - 8
- Simplify: x = 3
- Check: 4(3 + 2) = 4(5) = 20. And 3(3) + 11 = 9 + 11 = 20. Correct!
Step 1: 5x - 2x = 9 + 3 → 3x = 12
Step 2: x = 4
Check: 5(4) - 3 = 17 and 2(4) + 9 = 17. Correct!
Solving Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. Three main methods to solve it:
Method 1: Factoring
Best when the equation factors cleanly.
- x² - 5x + 6 = 0
- Factor: (x - 2)(x - 3) = 0
- Solutions: x = 2 or x = 3
Method 2: Quadratic Formula
Works for every quadratic equation:
x = (-b ± sqrt(b² - 4ac)) / 2a
- For x² + 2x - 8 = 0: a = 1, b = 2, c = -8
- x = (-2 ± sqrt(4 + 32)) / 2 = (-2 ± 6) / 2
- x = 2 or x = -4
Method 3: Completing the Square
Rewrite the equation so one side is a perfect square trinomial. This method also proves the quadratic formula.
Equivalent Fractions in Algebra
Understanding equivalent fractions is essential for algebra - they appear everywhere from solving equations to simplifying rational expressions. Two fractions are equivalent when they represent the same value, even though they look different.
How to Find Equivalent Fractions
Multiply both numerator and denominator by the same number:
- 2/3 = 4/6 (multiply by 2) = 6/9 (multiply by 3) = 10/15 (multiply by 5)
Divide both by their greatest common factor (GCF) to simplify:
- 12/18 = 2/3 (divide both by 6)
Common Equivalent Fractions Reference
| Original | x2 | x3 | x4 | Decimal |
|---|---|---|---|---|
| 1/2 | 2/4 | 3/6 | 4/8 | 0.5 |
| 1/3 | 2/6 | 3/9 | 4/12 | 0.333... |
| 2/3 | 4/6 | 6/9 | 8/12 | 0.666... |
| 3/4 | 6/8 | 9/12 | 12/16 | 0.75 |
| 5/6 | 10/12 | 15/18 | 20/24 | 0.833... |
Explore More Equivalent Fractions
More Algebra Topics
Beyond basic equations, algebra covers many interconnected topics. Here are the key areas you should know:
Algebraic Expressions
Learn the building blocks of algebra - terms, coefficients, variables, and how to simplify expressions by combining like terms and using the distributive property.
- Types: monomial, binomial, trinomial, polynomial
- Operations: add, subtract, multiply, divide
- FOIL method for multiplying binomials
Algebraic Fractions
Fractions with variables in the numerator or denominator. Also called rational expressions. Master simplifying, adding, subtracting, multiplying, and dividing them.
- Simplify by factoring and canceling
- Add/subtract with common denominators
- Watch for restricted values (denominator ≠ 0)
Factoring Polynomials
Factoring is the reverse of expanding. Learn to break down polynomials into simpler factors - essential for solving equations and simplifying expressions.
- GCF (Greatest Common Factor)
- Difference of squares: a² - b² = (a+b)(a-b)
- Trinomial factoring: x² + bx + c
- Grouping method for four terms
Dividing Polynomials
Divide polynomials using long division (works for any divisor) or synthetic division (fast shortcut for x - a divisors). Both give a quotient and remainder.
- Long division: 5-step repeat process
- Synthetic division: coefficients only, much faster
- Use 0 placeholders for missing terms
Remainder Theorem
A shortcut for finding remainders without doing long division. When P(x) is divided by (x - a), the remainder equals P(a). Just plug in and evaluate!
- Remainder Theorem: remainder = P(a)
- Factor Theorem: if P(a) = 0, then (x - a) is a factor
- Great for testing potential factors quickly
Polynomial Operations
Add, subtract, multiply, and divide polynomials. Learn about degree, leading coefficient, and standard form.
- Add/subtract: combine like terms
- Multiply: use distributive property or FOIL
- Understand degree and classification
Algebra Problem-Solving Strategies
When you're stuck on an algebra problem, try these approaches:
- Draw a picture: Visualize the problem, especially for word problems involving geometry or distance.
- Work backward: Start from what you know and reverse the steps.
- Substitute numbers: Try plugging in simple numbers (like 1, 2, or 0) to test if your approach works.
- Check your answer: Always substitute your solution back into the original equation to verify.
- Break it into smaller parts: Complex problems become manageable when you solve one piece at a time.
Let width = w. Then length = 2w + 3.
Perimeter: 2(w) + 2(2w + 3) = 36
2w + 4w + 6 = 36 → 6w = 30 → w = 5 cm
Length = 2(5) + 3 = 13 cm
Check: 2(5) + 2(13) = 10 + 26 = 36. Correct!
Common Mistakes to Avoid
- Forgetting to distribute: 2(x + 3) is 2x + 6, not 2x + 3.
- Sign errors: When moving terms across the equals sign, flip the sign. -3x = 12 means x = -4.
- Dividing by zero: Never divide both sides by a variable that could be zero. Always check.
- Cancelling terms vs. factors: In (x + 3)/(x + 5), you cannot cancel x. Only common factors can be cancelled.
Try our Solve for X Calculator, Quadratic Equation Solver, or Factoring Calculator for practice.