Example 1, Combine like terms
Simplify 3x + 5 + 2x − 7
5x − 2
- 1
Group like terms
(3x + 2x) + (5 − 7) - 2
Combine
5x + (−2) - 3
Result
5x − 2
Group x-terms together and constants together.
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Three Algebra Patterns
a² − b² = (a+b)(a−b) · (a+b)² = a² + 2ab + b²
Difference of squares and perfect square trinomial, the two most-used factoring patterns. Memorize them; they appear constantly.
Algebra is the art of manipulating expressions while preserving their value. The core moves are: combine like terms, distribute over parentheses, factor (the reverse of distribution), expand products, simplify fractions, and solve equations.
Factoring is the most subtle move, given a polynomial, rewrite it as a product. Common techniques: greatest common factor first, then look for difference of squares (a² − b²), perfect squares (a² ± 2ab + b²), trinomial pattern (find two numbers that multiply/add correctly), or grouping for four-term polynomials.
Worked Examples
Five core algebra moves, demonstrated
From combining like terms to factor by grouping, the five techniques that cover most algebra homework.
Example 2, Distribute
Simplify 3(x + 4) − 2(x − 1)
x + 14
- 1
Distribute
3x + 12 − 2x + 2 - 2
Combine
(3x − 2x) + (12 + 2) - 3
Result
x + 14
Distribute first, then combine like terms.
Example 3, Factor a quadratic
Factor x² + 5x + 6
(x + 2)(x + 3)
- 1
Identify product/sum
Need ab = 6 and a + b = 5 - 2
Find the pair
a = 2, b = 3 (since 2·3 = 6 and 2 + 3 = 5) - 3
Write as product
(x + 2)(x + 3)
Find two numbers that multiply to 6 and add to 5: 2 and 3.
Example 4, Difference of squares
Factor x² − 16
(x + 4)(x − 4)
- 1
Identify pattern
x² − 16 = x² − 4² - 2
Apply formula
(x + 4)(x − 4)
Pattern: a² − b² = (a + b)(a − b).
Example 5, Factor by grouping
Factor x³ + 2x² + 3x + 6
(x + 2)(x² + 3)
- 1
Group
(x³ + 2x²) + (3x + 6) - 2
Factor each group
x²(x + 2) + 3(x + 2) - 3
Common factor
(x + 2)(x² + 3)
Pair terms and look for a common factor.
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Algebra is the foundation of higher math, mastery here pays off in calculus, statistics, physics, and beyond. This solver covers simplification, distribution, factoring, and equation solving with the complete work shown. Each worked example demonstrates a different core technique so you can recognize patterns in your own problems.
Combining like terms
Terms with the same variable raised to the same power can be added or subtracted by combining their coefficients. 3x + 5x = 8x, but 3x + 5x² cannot be combined further. Constants combine with constants. This is the first step in nearly every algebraic simplification.
Distributing and expanding
Distribution is the rule a(b + c) = ab + ac. Use it to clear parentheses before combining like terms. When distributing a negative, every sign inside the parentheses flips. For products of two binomials like (x + 2)(x + 3), use FOIL (First, Outer, Inner, Last) or the general distributive property.
Factoring polynomials
Factoring undoes distribution, rewrite a sum as a product. Always check first for a greatest common factor across all terms. Then identify special patterns: difference of squares, perfect square trinomials, sum/difference of cubes. For general trinomials ax² + bx + c, find two numbers that multiply to ac and add to b.
When to use this solver
Use the calculator above to evaluate numerical expressions step by step. For symbolic algebra, like factoring a specific polynomial or simplifying a long expression, the worked examples above demonstrate the methodology. For instant symbolic algebra on any problem, use the photo solver on the homepage.
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