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81 results for “algebra”

Formulae (76)

Quadratic Formula

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

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

AlgebraIntermediate

Discriminant

\Delta = b^2 - 4ac

Determines the nature of the roots of a quadratic equation.

AlgebraIntermediate

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.

AlgebraIntermediate

Logarithm Product Rule

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

The logarithm of a product equals the sum of logarithms.

AlgebraIntermediate

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.

AlgebraIntermediate

Logarithm Power Rule

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

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

AlgebraIntermediate

Change of Base Formula

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

Converts a logarithm from one base to another.

AlgebraBasic

Difference of Squares

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

Factors the difference of two perfect squares.

AlgebraBasic

Perfect Square Trinomial

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

Expands the square of a binomial.

AlgebraIntermediate

Sum of Cubes

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

Factors the sum of two cubes.

AlgebraIntermediate

Difference of Cubes

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

Factors the difference of two cubes.

AlgebraIntermediate

Cube of a Binomial

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

Expands the cube of a binomial expression.

AlgebraBasic

Arithmetic Series Sum

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

Sum of the first n terms of an arithmetic sequence.

AlgebraBasic

Arithmetic nth Term

a_n = a_1 + (n-1)d

The nth term of an arithmetic sequence.

AlgebraIntermediate

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).

AlgebraIntermediate

Infinite Geometric Series

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

Sum of an infinite geometric series that converges.

AlgebraBasic

Product of Powers

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

Multiplying powers with the same base adds the exponents.

AlgebraBasic

Power of a Power

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

Raising a power to a power multiplies the exponents.

AlgebraIntermediate

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.

AlgebraIntermediate

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.

AlgebraBasic

FOIL Method

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

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

AlgebraBasic

Quotient of Powers

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

Dividing powers with the same base subtracts exponents.

AlgebraBasic

Zero Exponent

a^0 = 1

Any nonzero number to the power zero equals one.

AlgebraBasic

Negative Exponent

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

A negative exponent means reciprocal.

AlgebraIntermediate

Fractional Exponent

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

A fractional exponent is a root.

AlgebraBasic

Absolute Value

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

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

AlgebraBasic

Slope-Intercept Form

y = mx + b

Equation of a straight line.

AlgebraIntermediate

Point-Slope Form

y - y_1 = m(x - x_1)

Line through a point with a given slope.

AlgebraBasic

Direct Variation

y = kx

y changes directly in proportion to x.

AlgebraIntermediate

Inverse Variation

y = \frac{k}{x}

y changes inversely with x.

AlgebraAdvanced

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.

AlgebraIntermediate

Exponential Growth / Decay

A = A_0 e^{kt}

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

AlgebraIntermediate

Sum of Cubes

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

Factoring formula for a sum of two cubes.

AlgebraIntermediate

Difference of Cubes

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

Factoring formula for a difference of two cubes.

AlgebraIntermediate

Cube of a Binomial

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

Expansion of a binomial raised to the third power.

AlgebraIntermediate

Logarithm Product Rule

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

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

AlgebraIntermediate

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.

AlgebraIntermediate

Logarithm Power Rule

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

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

AlgebraAdvanced

Change of Base

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

Convert a logarithm to a different base.

AlgebraBasic

Absolute Value Property

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

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

AlgebraIntermediate

Geometric Mean

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

The nth root of the product of n numbers.

AlgebraAdvanced

Harmonic Mean

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

Reciprocal of the average of reciprocals.

AlgebraIntermediate

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.

AlgebraIntermediate

Parabola Vertex

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

x-coordinate of the vertex of a quadratic.

AlgebraAdvanced

Infinite Geometric Series

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

Sum of an infinite geometric series that converges.

AlgebraIntermediate

Finite Geometric Series

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

Sum of the first n terms of a geometric series.

AlgebraIntermediate

Rational Exponent

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

A fractional exponent is a root and a power combined.

AlgebraBasic

Negative Exponent

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

A negative exponent gives the reciprocal.

AlgebraBasic

Zero Exponent

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

Any nonzero number to the power zero is one.

AlgebraBasic

Quotient of Powers

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

Divide powers with the same base by subtracting exponents.

AlgebraAdvanced

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.

AlgebraAdvanced

Partial Fraction Setup

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

Decompose a rational expression into simpler fractions.

Linear AlgebraIntermediate

Determinant (2×2)

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Determinant of a 2×2 matrix.

Linear AlgebraIntermediate

Dot Product

\vec{a}\cdot\vec{b} = \sum_i a_i b_i = |a||b|\cos\theta

Scalar product of two vectors.

Linear AlgebraAdvanced

Cross Product Magnitude

|\vec{a}\times\vec{b}| = |a||b|\sin\theta

Magnitude of the vector perpendicular to two vectors.

Linear AlgebraIntermediate

Vector Magnitude

|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

Length of a vector.

Linear AlgebraAdvanced

Eigenvalue Equation

A\vec{v} = \lambda\vec{v}

Defines eigenvalues and eigenvectors of a matrix.

Linear AlgebraAdvanced

Inverse of a 2×2 Matrix

A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Inverse of a 2×2 matrix (ad−bc ≠ 0).

Linear AlgebraIntermediate

Trace of a Matrix

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraAdvanced

Vector Projection

\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

Projection of vector a onto vector b.

Linear AlgebraBasic

Matrix Addition

(A+B)_{ij} = a_{ij} + b_{ij}

Matrices of equal size add entry by entry.

Linear AlgebraBasic

Scalar Multiplication

(cA)_{ij} = c\,a_{ij}

Multiply every entry of a matrix by a scalar.

Linear AlgebraBasic

Identity Matrix

AI = IA = A

The identity matrix leaves any matrix unchanged under multiplication.

Linear AlgebraIntermediate

Matrix Transpose

(A^T)_{ij} = a_{ji}

Reflects a matrix across its main diagonal.

Linear AlgebraAdvanced

Determinant (3×3)

\det = a(ei-fh) - b(di-fg) + c(dh-eg)

Determinant of a 3×3 matrix by cofactor expansion.

Linear AlgebraIntermediate

Matrix Trace

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraBasic

Dot Product

\mathbf{a}\cdot\mathbf{b} = \sum a_i b_i

Sum of componentwise products of two vectors.

Linear AlgebraBasic

Vector Magnitude

\|\mathbf{v}\| = \sqrt{\sum v_i^2}

Length of a vector.

Linear AlgebraIntermediate

Angle Between Vectors

\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|a\|\|b\|}

Angle from the dot product and magnitudes.

Linear AlgebraAdvanced

Scalar Triple Product

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})

Volume of the parallelepiped spanned by three vectors.

Linear AlgebraBasic

Identity Matrix Property

AI = IA = A

Multiplying by the identity leaves a matrix unchanged.

Linear AlgebraAdvanced

Transpose of a Product

(AB)^T = B^T A^T

The transpose of a product reverses the order.

Linear AlgebraAdvanced

Rank-Nullity Theorem

\text{rank}(A) + \text{nullity}(A) = n

Relates rank and null space dimension to columns.

Linear AlgebraIntermediate

2×2 Matrix Product

\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix} = \begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}

Product of two 2×2 matrices.

Linear AlgebraBasic

Unit Vector

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

Vector of length one in the same direction.

Linear AlgebraAdvanced

Determinant of a Product

\det(AB) = \det(A)\det(B)

The determinant of a product is the product of determinants.

Math Tricks (5)

Sum of 1 to N Instantly

Gauss's formula: the sum of the first n whole numbers is n(n+1)/2 — no adding required.

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Sum of First N Odd Numbers = N²

The sum of the first n odd numbers is always a perfect square, exactly n².

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Difference of Squares Shortcut

a² − b² = (a + b)(a − b). Turns a subtraction of squares into an easy product.

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FOIL for Binomial Products

Multiply two binomials by combining First, Outer, Inner, Last terms.

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Product of Numbers Around a Square

Two numbers equally spaced around a middle value m multiply to m² − d², using the difference of squares.

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