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Algebra–Formulas and Definitions

Real-Number Properties

$\begin{array}{l} a + b = b + a & a b = b a \\ (a + b) + c = a + (b + c) & (a b) c = a (b c) \\ a (b + c) = a b + a c & (a + b) c = a c + b c \\ a + 0 = 0 + a = a & a \cdot 1 = 1 \cdot a = a \\ a + (- a) = 0 & a (\frac{1}{a}) = 1, a \neq 0 \\ a \cdot 0 = 0 \cdot a = 0 \end{array}$ $\begin{array}{l} a + b = b + a & a b = b a \\ (a + b) + c = a + (b + c) & (a b) c = a (b c) \\ a (b + c) = a b + a c & (a + b) c = a c + b c \\ a + 0 = 0 + a = a & a \cdot 1 = 1 \cdot a = a \\ a + (- a) = 0 & a (\frac{1}{a}) = 1, a \neq 0 \\ a \cdot 0 = 0 \cdot a = 0 \end{array}$

If $a \cdot b = 0$ $a \cdot b = 0$ , then $a = 0$ $a = 0$ or $b = 0$ $b = 0$ .

Absolute Value

$| a | = a$ $| a | = a$ if $a \geq 0$ $a \geq 0$ and $| a | = - a$ $| a | = - a$ if $a < 0$ $a < 0$ .

Assume that $a > 0$ $a > 0$ and u is an algebraic expression:

$| u | = a$ $| u | = a$ is equivalent to $u = a$ $u = a$ or $u = - a$ $u = - a$ .

$| u | < a$ $| u | < a$ is equivalent to $- a < u < a$ $- a < u < a$ .

$| u | > a$ $| u | > a$ is equivalent to $u < - a$ $u < - a$ or $u > a$ $u > a$ .

Exponents

$\begin{array}{l} a^{n} = \underset{n factor}{\underset{︸}{a \cdot a \dots a}} & a^{0} = 1, a \neq 0 \\ a^{- n} = \frac{1}{a^{n}}, a \neq 0 \\ a^{m} a^{n} = a^{m + n} & \frac{a^{m}}{a^{n}} = a^{m - n} \\ {(a^{m})}^{n} = a^{m n} & {(a \cdot b)}^{n} = a^{n} \cdot b^{n} \\ {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}} & {(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n} = \frac{b^{n}}{a^{n}} \end{array}$ $\begin{array}{l} a^{n} = \underset{n factor}{\underset{︸}{a \cdot a \dots a}} & a^{0} = 1, a \neq 0 \\ a^{- n} = \frac{1}{a^{n}}, a \neq 0 \\ a^{m} a^{n} = a^{m + n} & \frac{a^{m}}{a^{n}} = a^{m - n} \\ {(a^{m})}^{n} = a^{m n} & {(a \cdot b)}^{n} = a^{n} \cdot b^{n} \\ {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}} & {(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n} = \frac{b^{n}}{a^{n}} \end{array}$

Logarithms

$\begin{array}{l} \begin{array}{l} y = \log_{a} x means a^{y} = x \\ \log_{a} a^{x} = x & a^{\log_{a} x} = x \\ \log_{a} 1 = 0 & \log_{a} a = 1 \\ \log x = \log_{10} x & ln x = \log_{e} x \end{array} \\ \begin{array}{l} \log_{a} M N = \log_{a} M + \log_{a} N \\ \log_{a} \frac{M}{N} = \log_{a} M - \log_{a} N \\ \log_{a} M^{r} = r \log_{a} M \\ \log_{b} x = \frac{\log_{a} x}{\log_{a} b} = \frac{\log x}{\log b} = \frac{ln x}{ln b} (change-of-base formula) \end{array} \end{array}$ $\begin{array}{l} \begin{array}{l} y = \log_{a} x means a^{y} = x \\ \log_{a} a^{x} = x & a^{\log_{a} x} = x \\ \log_{a} 1 = 0 & \log_{a} a = 1 \\ \log x = \log_{10} x & ln x = \log_{e} x \end{array} \\ \begin{array}{l} \log_{a} M N = \log_{a} M + \log_{a} N \\ \log_{a} \frac{M}{N} = \log_{a} M - \log_{a} N \\ \log_{a} M^{r} = r \log_{a} M \\ \log_{b} x = \frac{\log_{a} x}{\log_{a} b} = \frac{\log x}{\log b} = \frac{ln x}{ln b} (change-of-base formula) \end{array} \end{array}$

Radicals

$\begin{array}{l} \sqrt{a^{2}} = | a | \\ \sqrt[n]{a^{n}} = | a | if n is even \\ \sqrt[n]{a^{n}} = a if n is odd \\ a^{\frac{1}{n}} = \sqrt{n a} & a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} = \sqrt[n]{a^{m}} \\ \sqrt[n]{a b} = \sqrt[n]{a} \sqrt[n]{b} & \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{array}$ $\begin{array}{l} \sqrt{a^{2}} = | a | \\ \sqrt[n]{a^{n}} = | a | if n is even \\ \sqrt[n]{a^{n}} = a if n is odd \\ a^{\frac{1}{n}} = \sqrt{n a} & a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} = \sqrt[n]{a^{m}} \\ \sqrt[n]{a b} = \sqrt[n]{a} \sqrt[n]{b} & \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{array}$

Rational Expressions

(Assume all denominators not to be zero.)

$\begin{array}{l} \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} \\ \frac{\frac{A}{B}}{\frac{C}{D}} = \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{A \cdot D}{B \cdot C} \\ \frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C} & \frac{A}{B} \pm \frac{C}{D} = \frac{A \cdot D \pm B \cdot C}{B \cdot D} \end{array}$ $\begin{array}{l} \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} \\ \frac{\frac{A}{B}}{\frac{C}{D}} = \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{A \cdot D}{B \cdot C} \\ \frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C} & \frac{A}{B} \pm \frac{C}{D} = \frac{A \cdot D \pm B \cdot C}{B \cdot D} \end{array}$

Common Products and Factors

$\begin{array}{l} A^{2} - B^{2} = (A + B) (A - B) \\ A^{2} + 2 A B + B^{2} = {(A + B)}^{2} \\ A^{2} - 2 A B + B^{2} = {(A - B)}^{2} \\ A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2}) \\ A^{3} + B^{3} = (A + B) (A^{2} - A B + B^{2}) \\ A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3} = {(A + B)}^{3} \\ A^{3} - 3 A^{2} B + 3 A B^{2} - B^{3} = {(A - B)}^{3} \end{array}$ $\begin{array}{l} A^{2} - B^{2} = (A + B) (A - B) \\ A^{2} + 2 A B + B^{2} = {(A + B)}^{2} \\ A^{2} - 2 A B + B^{2} = {(A - B)}^{2} \\ A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2}) \\ A^{3} + B^{3} = (A + B) (A^{2} - A B + B^{2}) \\ A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3} = {(A + B)}^{3} \\ A^{3} - 3 A^{2} B + 3 A B^{2} - B^{3} = {(A - B)}^{3} \end{array}$

Distance Formula

The distance between $P (x_{1}, y_{1})$ $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ $Q (x_{2}, y_{2})$ is

d (P, Q) = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .

$d (P, Q) = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} .$

Midpoint Formula

The midpoint M(x, y) of the line segment with endpoints $P (x_{1}, y_{1})$ $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ $Q (x_{2}, y_{2})$ is

M (x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .

$M (x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .$

Lines

Slope $m = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $m = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Point-slope form	$y - y_{1} = m (x - x_{1})$ $y - y_{1} = m (x - x_{1})$
Slope-intercept form	$y = m x + b$ $y = m x + b$
Horizontal line	$y = k$ $y = k$
Vertical line	$x = k$ $x = k$
General form	$a x + b y + c = 0$ $a x + b y + c = 0$

Quadratic Formula

The solutions to the equation $a x^{2} + b x + c = 0, a \neq 0$ $a x^{2} + b x + c = 0, a \neq 0$ , are $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

Standard Form of a Circle

$\begin{array}{l} {(x - h)}^{2} + {(y - k)}^{2} = r^{2} & \begin{array}{l} center : (h, k) \\ radius: r \end{array} \end{array}$ $\begin{array}{l} {(x - h)}^{2} + {(y - k)}^{2} = r^{2} & \begin{array}{l} center : (h, k) \\ radius: r \end{array} \end{array}$

Standard Form of a Parabola

$\begin{array}{rcll} {(y - k)}^{2} & = & \pm 4 p (x - h) & \begin{array}{l} vertex : (h, k) \\ and (p > 0) \end{array} \\ {(x - h)}^{2} & = & \pm 4 p (y - k) \end{array}$ $\begin{array}{rcll} {(y - k)}^{2} & = & \pm 4 p (x - h) & \begin{array}{l} vertex : (h, k) \\ and (p > 0) \end{array} \\ {(x - h)}^{2} & = & \pm 4 p (y - k) \end{array}$

Standard Form of an Ellipse

$\begin{array}{l} \frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1; \\ a > b > 0 (Horizontal ellipse) \\ \frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1; \\ a > b > 0 (Vertical ellipse) \end{array}$ $\begin{array}{l} \frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1; \\ a > b > 0 (Horizontal ellipse) \\ \frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1; \\ a > b > 0 (Vertical ellipse) \end{array}$

Standard Form of a Hyperbola

$\begin{array}{l} \frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1; \\ a > 0, b > 0 (Horizontal) \\ \frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1; \\ a > 0, b > 0 (Vertical) \end{array}$ $\begin{array}{l} \frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1; \\ a > 0, b > 0 (Horizontal) \\ \frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1; \\ a > 0, b > 0 (Vertical) \end{array}$