Algebra

Equations, polynomials, identities and algebraic laws

Algebra brings together the essential formulas for working with equations, polynomials, identities, and key algebraic laws, from the Quadratic Formula and Binomial Theorem to logarithm rules and common factoring patterns. It’s a practical reference for students, exam prep, and anyone using maths in real-world problem solving, because these formulas make it faster to simplify expressions, solve unknowns, and spot patterns with confidence.

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds the roots of any quadratic equation ax² + bx + c = 0.

Discriminant

\Delta = b^2 - 4ac

Determines the nature of the roots of a quadratic equation.

Binomial Theorem

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expands any power of a binomial into a sum of terms.

Logarithm Product Rule

\log_b(xy) = \log_b(x) + \log_b(y)

The logarithm of a product equals the sum of logarithms.

Logarithm Quotient Rule

\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

The logarithm of a quotient equals the difference of logarithms.

Logarithm Power Rule

\log_b(x^n) = n \cdot \log_b(x)

The logarithm of a power equals the exponent times the logarithm.

Change of Base Formula

\log_b(x) = \frac{\log_k(x)}{\log_k(b)}

Converts a logarithm from one base to another.

Difference of Squares

a^2 - b^2 = (a+b)(a-b)

Factors the difference of two perfect squares.

Perfect Square Trinomial

(a \pm b)^2 = a^2 \pm 2ab + b^2

Expands the square of a binomial.

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factors the sum of two cubes.

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factors the difference of two cubes.

Cube of a Binomial

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expands the cube of a binomial expression.

Arithmetic Series Sum

S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence.

Arithmetic nth Term

a_n = a_1 + (n-1)d

The nth term of an arithmetic sequence.

Geometric Series Sum

S_n = a_1 \frac{1 - r^n}{1 - r}

Sum of the first n terms of a geometric sequence (r ≠ 1).

Infinite Geometric Series

S = \frac{a_1}{1 - r}, \quad |r| < 1

Sum of an infinite geometric series that converges.

Product of Powers

a^m \cdot a^n = a^{m+n}

Multiplying powers with the same base adds the exponents.

Power of a Power

(a^m)^n = a^{mn}

Raising a power to a power multiplies the exponents.

Completing the Square

ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Rewrites a quadratic in vertex form.

Vieta's Formulas

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Relates the roots of a quadratic to its coefficients.

FOIL Method

(a+b)(c+d) = ac + ad + bc + bd

Multiply two binomials: First, Outer, Inner, Last.

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Dividing powers with the same base subtracts exponents.

Zero Exponent

a^0 = 1

Any nonzero number to the power zero equals one.

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal.

Fractional Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root.

Absolute Value

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Distance of a number from zero, always non-negative.

Slope-Intercept Form

y = mx + b

Equation of a straight line.

Point-Slope Form

y - y_1 = m(x - x_1)

Line through a point with a given slope.

Direct Variation

y = kx

y changes directly in proportion to x.

Inverse Variation

y = \frac{k}{x}

y changes inversely with x.

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

Closed form for the sum of the first n cubes.

Exponential Growth / Decay

A = A_0 e^{kt}

Growth (k>0) or decay (k<0) over time.

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factoring formula for a sum of two cubes.

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factoring formula for a difference of two cubes.

Cube of a Binomial

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expansion of a binomial raised to the third power.

Logarithm Product Rule

\log_b(xy) = \log_b x + \log_b y

The log of a product is the sum of the logs.

Logarithm Quotient Rule

\log_b\frac{x}{y} = \log_b x - \log_b y

The log of a quotient is the difference of the logs.

Logarithm Power Rule

\log_b(x^n) = n\log_b x

The log of a power brings the exponent to the front.

Change of Base

\log_b x = \frac{\log_k x}{\log_k b}

Convert a logarithm to a different base.

Absolute Value Property

|ab| = |a|\,|b|

The absolute value of a product equals the product of absolute values.

Geometric Mean

GM = \sqrt[n]{x_1 x_2 \cdots x_n}

The nth root of the product of n numbers.

Harmonic Mean

HM = \frac{n}{\sum 1/x_i}

Reciprocal of the average of reciprocals.

Completing the Square

x^2 + bx = \left(x+\frac{b}{2}\right)^2 - \frac{b^2}{4}

Rewrite a quadratic as a perfect square plus a constant.

Parabola Vertex

x = -\frac{b}{2a}

x-coordinate of the vertex of a quadratic.

Infinite Geometric Series

S = \frac{a}{1-r}, \; |r|<1

Sum of an infinite geometric series that converges.

Finite Geometric Series

S_n = a\frac{1-r^n}{1-r}

Sum of the first n terms of a geometric series.

Rational Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root and a power combined.

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent gives the reciprocal.

Zero Exponent

a^0 = 1, \; a\neq 0

Any nonzero number to the power zero is one.

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Divide powers with the same base by subtracting exponents.

System of Two Linear Equations

x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

Cramer-rule solution for two simultaneous equations.

Partial Fraction Setup

\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decompose a rational expression into simpler fractions.