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Algebra Formulas with Worked Examples

Review algebra formulas for linear equations, quadratic roots, exponents and factoring, with conditions and worked examples.

Algebra formula reference

Topic	Formula and conditions
Linear equation	`ax + b = c → x = (c − b)/a` a ≠ 0. If a = 0, compare b and c instead of dividing by zero.
Distributive law	`a(b + c) = ab + ac` Multiply a by every term inside the brackets.
Quadratic formula	`x = (−b ± √(b^2 − 4ac))/(2a)` For ax^2 + bx + c = 0, with a ≠ 0.
Discriminant	`D = b^2 − 4ac` D > 0: two real roots; D = 0: repeated root; D < 0: complex pair.
Square of a sum	`(a + b)^2 = a^2 + 2ab + b^2` The middle term is 2ab; do not omit it.
Square of a difference	`(a − b)^2 = a^2 − 2ab + b^2` The last term is positive.
Difference of squares	`a^2 − b^2 = (a − b)(a + b)` Both terms must be squares.
Sum of cubes	`a^3 + b^3 = (a + b)(a^2 − ab + b^2)` The quadratic factor has a negative middle term.
Difference of cubes	`a^3 − b^3 = (a − b)(a^2 + ab + b^2)` The quadratic factor has a positive middle term.
Product of powers	`a^m a^n = a^(m+n)` For nonnegative integer m and n. Negative exponents require a ≠ 0.
Quotient of powers	`a^m/a^n = a^(m−n)` a ≠ 0; division by zero is undefined.
Power of a power	`(a^m)^n = a^(mn)` Use integer exponent rules here; fractional powers require care with domains.
Two linear equations	`D = ad − bc; x = (ed − bf)/D; y = (af − ec)/D` For ax + by = e and cx + dy = f, when D ≠ 0.
Adding fractions	`A/B + C/D = (AD + BC)/(BD)` B ≠ 0 and D ≠ 0. Cancel factors only after combining.
Dividing fractions	`(A/B) ÷ (C/D) = AD/(BC)` B ≠ 0, D ≠ 0 and C ≠ 0. The divisor must not be zero.

Choose a formula by the structure of the problem

A formula is useful only when its conditions match the input. The quadratic formula needs a nonzero coefficient of x². A fraction operation requires nonzero denominators. An exponent rule used for whole-number powers should not be extended to arbitrary fractional powers without checking the domain.

Write the expression clearly before substituting numbers. For example, the denominator in the quadratic formula is the whole quantity 2a, so enter it as (2*a). If you enter a negative value for b, use parentheses when squaring it: (−5)^2 = 25, while −5^2 = −25.

Worked formula examples

Linear equation

For 4x + 7 = 31, identify a = 4, b = 7 and c = 31. Then x = (31 − 7)/4 = 6. Checking gives 4 × 6 + 7 = 31.

Solve this example

Quadratic equation

For x^2 − 6x + 8 = 0, a = 1, b = −6 and c = 8. The discriminant is 36 − 32 = 4. The roots are (6 ± 2)/2, giving x = 4 and x = 2. Both make the original expression zero.

See the quadratic working

Difference of squares

9x^2 − 25 is (3x)^2 − 5^2, so the factors are (3x − 5)(3x + 5). Expanding cancels the middle terms and recovers 9x² − 25.

Factor this example

Common formula mistakes

The square of a sum includes a middle term: (x + 3)² is x² + 6x + 9. Cancelling an x from (x + 2)/x is invalid because x is a term in the numerator, not a factor of the entire numerator. Dividing both sides of an equation by a variable can also lose a solution if that variable could be zero.

Record restrictions first, apply one justified operation at a time, and check the final answer in the original expression or equation. The identity checker can verify polynomial formulas; the fractions tool keeps original exclusions visible.

Further reading: OpenStax: polynomials and rational expressions.