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Algebra

Equations, polynomials, identities and algebraic laws

Quadratic Formula

Intermediate
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

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Discriminant

Intermediate
Δ=b2−4ac\Delta = b^2 - 4ac

Determines the nature of the roots of a quadratic equation.

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Binomial Theorem

Intermediate
(a+b)n=∑k=0n(nk)an−kbk(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.

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Logarithm Product Rule

Intermediate
log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)

The logarithm of a product equals the sum of logarithms.

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Logarithm Quotient Rule

Intermediate
log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

The logarithm of a quotient equals the difference of logarithms.

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Logarithm Power Rule

Intermediate
log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)

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

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Change of Base Formula

Intermediate
log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}

Converts a logarithm from one base to another.

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Difference of Squares

Basic
a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)

Factors the difference of two perfect squares.

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Perfect Square Trinomial

Basic
(a±b)2=a2±2ab+b2(a \pm b)^2 = a^2 \pm 2ab + b^2

Expands the square of a binomial.

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Sum of Cubes

Intermediate
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factors the sum of two cubes.

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Difference of Cubes

Intermediate
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factors the difference of two cubes.

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Cube of a Binomial

Intermediate
(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expands the cube of a binomial expression.

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Arithmetic Series Sum

Basic
Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence.

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Arithmetic nth Term

Basic
an=a1+(n−1)da_n = a_1 + (n-1)d

The nth term of an arithmetic sequence.

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Geometric Series Sum

Intermediate
Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}

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

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Infinite Geometric Series

Intermediate
S=a11−r,∣r∣<1S = \frac{a_1}{1 - r}, \quad |r| < 1

Sum of an infinite geometric series that converges.

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Product of Powers

Basic
am⋅an=am+na^m \cdot a^n = a^{m+n}

Multiplying powers with the same base adds the exponents.

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Power of a Power

Basic
(am)n=amn(a^m)^n = a^{mn}

Raising a power to a power multiplies the exponents.

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Completing the Square

Intermediate
ax2+bx+c=a(x+b2a)2+c−b24aax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

Rewrites a quadratic in vertex form.

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Vieta's Formulas

Intermediate
x1+x2=−ba,x1x2=cax_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Relates the roots of a quadratic to its coefficients.

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FOIL Method

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

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

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Quotient of Powers

Basic
aman=am−n\frac{a^m}{a^n} = a^{m-n}

Dividing powers with the same base subtracts exponents.

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Zero Exponent

Basic
a0=1a^0 = 1

Any nonzero number to the power zero equals one.

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Negative Exponent

Basic
a−n=1ana^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal.

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Fractional Exponent

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

A fractional exponent is a root.

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Absolute Value

Basic
∣x∣={xx≥0−xx<0|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

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

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Slope-Intercept Form

Basic
y=mx+by = mx + b

Equation of a straight line.

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Point-Slope Form

Intermediate
y−y1=m(x−x1)y - y_1 = m(x - x_1)

Line through a point with a given slope.

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Direct Variation

Basic
y=kxy = kx

y changes directly in proportion to x.

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Inverse Variation

Intermediate
y=kxy = \frac{k}{x}

y changes inversely with x.

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Sum of First n Cubes

Advanced
∑k=1nk3=(n(n+1)2)2\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

Closed form for the sum of the first n cubes.

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Exponential Growth / Decay

Intermediate
A=A0ektA = A_0 e^{kt}

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

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Sum of Cubes

Intermediate
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factoring formula for a sum of two cubes.

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Difference of Cubes

Intermediate
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factoring formula for a difference of two cubes.

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Cube of a Binomial

Intermediate
(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expansion of a binomial raised to the third power.

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Logarithm Product Rule

Intermediate
log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y

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

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Logarithm Quotient Rule

Intermediate
log⁡bxy=log⁡bx−log⁡by\log_b\frac{x}{y} = \log_b x - \log_b y

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

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Logarithm Power Rule

Intermediate
log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x

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

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Change of Base

Advanced
log⁡bx=log⁡kxlog⁡kb\log_b x = \frac{\log_k x}{\log_k b}

Convert a logarithm to a different base.

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Absolute Value Property

Basic
∣ab∣=∣a∣ ∣b∣|ab| = |a|\,|b|

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

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Geometric Mean

Intermediate
GM=x1x2⋯xnnGM = \sqrt[n]{x_1 x_2 \cdots x_n}

The nth root of the product of n numbers.

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Harmonic Mean

Advanced
HM=n∑1/xiHM = \frac{n}{\sum 1/x_i}

Reciprocal of the average of reciprocals.

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Completing the Square

Intermediate
x2+bx=(x+b2)2−b24x^2 + bx = \left(x+\frac{b}{2}\right)^2 - \frac{b^2}{4}

Rewrite a quadratic as a perfect square plus a constant.

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Parabola Vertex

Intermediate
x=−b2ax = -\frac{b}{2a}

x-coordinate of the vertex of a quadratic.

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Infinite Geometric Series

Advanced
S=a1−r,  ∣r∣<1S = \frac{a}{1-r}, \; |r|<1

Sum of an infinite geometric series that converges.

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Finite Geometric Series

Intermediate
Sn=a1−rn1−rS_n = a\frac{1-r^n}{1-r}

Sum of the first n terms of a geometric series.

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Rational Exponent

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

A fractional exponent is a root and a power combined.

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Negative Exponent

Basic
a−n=1ana^{-n} = \frac{1}{a^n}

A negative exponent gives the reciprocal.

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Zero Exponent

Basic
a0=1,  a≠0a^0 = 1, \; a\neq 0

Any nonzero number to the power zero is one.

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Quotient of Powers

Basic
aman=am−n\frac{a^m}{a^n} = a^{m-n}

Divide powers with the same base by subtracting exponents.

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System of Two Linear Equations

Advanced
x=c1b2−c2b1a1b2−a2b1x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

Cramer-rule solution for two simultaneous equations.

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Partial Fraction Setup

Advanced
1(x−a)(x−b)=Ax−a+Bx−b\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decompose a rational expression into simpler fractions.

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