Properties of Real Numbers

Commutative:

\[ \begin{aligned} a+b=b+a\\ a b=b a \end{aligned} \]

Associative:

\[ \begin{aligned} (a+b)+c&=a+(b+c)\\ (a b) c&=a(b c) \end{aligned} \]

Distributive: $$ a(b+c)=a b+a c $$

Absolute Value

\[ \begin{aligned} & |a|= \begin{cases}a & \text { if } a \geq 0 \\ -a & \text { if } a<0\end{cases} \\ & |a b|=|a||b| \\ & \left|\frac{a}{b}\right|=\frac{|a|}{|b|} \end{aligned} \]

Distance between \(a\) and \(b\) :

\[ d(a, b)=|b-a| \]

Exponents

\[ \begin{aligned} a^{m} a^{n}&=a^{m+n} \\ \frac{a^{m}}{a^{n}}&=a^{m-n} \\ \left(a^{m}\right)^{n}&=a^{m n} \\ (a b)^{n}&=a^{n} b^{n} \\ \left(\frac{a}{b}\right)^{n}&=\frac{a^{n}}{b^{n}} \end{aligned} \]

Radicals

\[ \sqrt[n]{a}=b \text{ means } b^{n}=a \]
\[ \sqrt[n]{a b}=\sqrt[n]{a} \sqrt[n]{b} \]
\[ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \]
\[ \sqrt[m]{\sqrt[n]{a}}=\sqrt[m n]{a} \]
\[ a^{m / n}=\sqrt[n]{a^{m}} \]

If \(n\) is odd, then \(\sqrt[n]{a^{n}}=a\). If \(n\) is even, then \(\sqrt[n]{a^{n}}=|a|\).

Special Product Formulas

Sum and difference of same terms:

\[ (A+B)(A-B)=A^{2}-B^{2} \]

Square of a sum or difference:

\[ \begin{aligned} & (A+B)^{2}=A^{2}+2 A B+B^{2} \\ & (A-B)^{2}=A^{2}-2 A B+B^{2} \end{aligned} \]

Cube of a sum or difference:

\[ \begin{aligned} & (A+B)^{3}=A^{3}+3 A^{2} B+3 A B^{2}+B^{3} \\ & (A-B)^{3}=A^{3}-3 A^{2} B+3 A B^{2}-B^{3} \end{aligned} \]

Special Factoring Formulas

Difference of squares:

\[ A^{2}-B^{2}=(A+B)(A-B) \]

Perfect squares:

\[ \begin{aligned} & A^{2}+2 A B+B^{2}=(A+B)^{2} \\ & A^{2}-2 A B+B^{2}=(A-B)^{2} \end{aligned} \]

Sum or difference of cubes:

\[ \begin{aligned} & A^{3}-B^{3}=(A-B)\left(A^{2}+A B+B^{2}\right) \\ & A^{3}+B^{3}=(A+B)\left(A^{2}-A B+B^{2}\right) \end{aligned} \]

Rational Expressions

We can cancel common factors:

\[ \frac{A C}{B C}=\frac{A}{B} \]

To multiply two fractions, we multiply their numerators together and their denominators together:

\[ \frac{A}{B} \times \frac{C}{D}=\frac{A C}{B D} \]

To divide fractions, we invert the divisor and multiply:

\[ \frac{A}{B} \div \frac{C}{D}=\frac{A}{B} \times \frac{D}{C} \]

To add fractions, we find a common denominator:

\[ \frac{A}{C}+\frac{B}{C}=\frac{A+B}{C} \]

Properties of Equality

\(A=B \quad \Leftrightarrow \quad A+C=B+C\) \(A=B \quad \Leftrightarrow \quad C A=C B \quad(C \neq 0)\)

Linear Equations

A linear equation in one variable is an equation of the form \(a x+b=0\).

Zero-Product Property

If \(A B=0\), then \(A=0\) or \(B=0\).

Completing the Square

To make \(x^{2}+b x\) a perfect square, add \(\left(\frac{b}{2}\right)^{2}\). This gives the perfect square

\[ x^{2}+b x+\left(\frac{b}{2}\right)^{2}=\left(x+\frac{b}{2}\right)^{2} \]

Quadratic Formula

A quadratic equation is an equation of the form

\[ a x^{2}+b x+c=0 \]

Its solutions are given by the Quadratic Formula:

\[ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \]

The discriminant is \(D=b^{2}-4 a c\). If \(D>0\), the equation has two real solutions. If \(D=0\), the equation has one solution. If \(D<0\), the equation has two complex solutions.

Complex Numbers

A complex number is a number of the form \(a+b i\), where \(i=\sqrt{-1}\).

The complex conjugate of \(a+b i\) is

\[ \overline{a+b i}=a-b i \]

To multiply complex numbers, treat them as binomials and use \(i^{2}=-1\) to simplify the result.

To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator:

\[ \frac{a+b i}{c+d i}=\left(\frac{a+b i}{c+d i}\right) \cdot\left(\frac{c-d i}{c-d i}\right)=\frac{(a+b i)(c-d i)}{c^{2}+d^{2}} \]

Inequalities

Adding the same quantity to each side of an inequality gives an equivalent inequality:

\[ A<B \quad \Leftrightarrow \quad A+C<B+C \]

Multiplying each side of an inequality by the same positive quantity gives an equivalent inequality. Multiplying each side by the same negative quantity reverses the direction of the inequality:

\[ \begin{aligned} & \text { If } C>0 \text {, then } A<B \quad \Leftrightarrow \quad C A<C B \\ & \text { If } C<0 \text {, then } A<B \quad \Leftrightarrow \quad C A>C B \end{aligned} \]

Absolute-Value Inequalities

To solve absolute-value inequalities, we use

\[ \begin{aligned} & |x|<C \quad \Leftrightarrow \quad-C<x<C \\ & |x|>C \quad \Leftrightarrow \quad x<-C \quad \text { or } x>C \end{aligned} \]

The Distance Formula

The distance between the points \(A\left(x_{1}, y_{1}\right)\) and \(B\left(x_{2}, y_{2}\right)\) is

\[ d(A, B)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \]

The Midpoint Formula

The midpoint of the line segment from \(A\left(x_{1}, y_{1}\right)\) to \(B\left(x_{2}, y_{2}\right)\) is

\[ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) \]

Intercepts

To find the \(x\)-intercepts of the graph of an equation, set \(y=0\) and solve for \(x\).

To find the \(y\)-intercepts of the graph of an equation, set \(x=0\) and solve for \(y\).

Circles

The circle with center \((0,0)\) and radius \(r\) has equation

\[ x^{2}+y^{2}=r^{2} \]

The circle with center \((h, k)\) and radius \(r\) has equation

\[ (x-h)^{2}+(y-k)^{2}=r^{2} \]

Symmetry

The graph of an equation is symmetric with respect to the \(\boldsymbol{x}\)-axis if the equation remains unchanged when \(y\) is replaced by \(-y\).

The graph of an equation is symmetric with respect to the \(y\)-axis if the equation remains unchanged when \(x\) is replaced by \(-x\). The graph of an equation is symmetric with respect to the origin if the equation remains unchanged when \(x\) is replaced by \(-x\) and \(y\) by \(-y\).

Slope of a Line

The slope of the nonvertical line that contains the points \(A\left(x_{1}, y_{1}\right)\) and \(B\left(x_{2}, y_{2}\right)\) is

\[ m=\frac{\text { rise }}{\text { run }}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \]

Equations of Lines

If a line has slope \(m\), has \(y\)-intercept \(b\), and contains the point \(\left(x_{1}, y_{1}\right)\), then: the point-slope form of its equation is

\[ y-y_{1}=m\left(x-x_{1}\right) \]

the slope-intercept form of its equation is

\[ y=m x+b \]

The equation of any line can be expressed in the general form

\[ A x+B y+C=0 \]

where \(A\) and \(B\) can't both be 0 .

Vertical and Horizontal Lines

The vertical line containing the point \((a, b)\) has the equation \(x=a\).

The horizontal line containing the point \((a, b)\) has the equation \(y=b\).

Parallel and Perpendicular Lines

Two lines with slopes \(m_{1}\) and \(m_{2}\) are parallel if and only if \(m_{1}=m_{2}\) perpendicular if and only if \(m_{1} m_{2}=-1\)

Variation

If \(y\) is directly proportional to \(x\), then

\[ y=k x \]

If \(y\) is inversely proportional to \(x\), then

\[ y=\frac{k}{x} \]