Algebra Cheat Sheet

Common Algebraic Errors

Arithmetic Operations

$\begin{split} ab+ac &= a \left(b+c \right)\\ \\ \frac{\left( \frac{a}{b} \right)}{c} &= \frac{a}{bc}\\ \\ \frac{a}{b} + \frac{c}{d} &= \frac{ad+bc}{bd}\\ \\ \frac{a-b}{c-d} &= \frac{b-a}{d-c}\\ \\ \frac{ab+ac}{a} &= b+c \text{, } a \neq 0\\ \\ a \left(\frac{b}{c} \right) &= \frac{ab}{c}\\ \\ \frac{a}{\left ( \frac{b}{c} \right )} &= \frac{ac}{b}\\ \\ \frac{a}{b} – \frac{c}{d} &= \frac{ad – bc}{bd}\\ \\ \frac{a+b}{c} &= \frac{a}{c} + \frac{b}{c}\\ \\ \frac{\left( \frac{a}{b} \right)}{\left( \frac{c}{d} \right)} &= \frac{ad}{bc} \end{split}$

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Exponent Properties

$\begin{split} a^{n}a^{m} &= a^{m+n} \\ \\ \left ( a^{n} \right )^{m} &= a^{nm} \\ \\ \left ( ab \right )^{n} &= a^{n}b^{n} \\ \\ a^{-n} &= \frac{1}{a^{n}} \\ \\ \left ( \frac{a}{b} \right )^{-n} &= \left ( \frac{b}{a} \right )^{n} = \frac{b^{n}}{a^{n}} \\ \\ \frac{a^{n}}{a^{m}} &= a^{n-m} = \frac{1}{a^{m-n}} \\ \\ a^{0} &= 1 \text{, } a \neq 0 \\ \\ \left ( \frac{a}{b} \right )^{n} &= \frac{a^{n}}{b^{n}} \\ \\ \frac{1}{a^{-n}} &= a^{n}\\ \\ a^{\frac{n}{m}} &= \left ( a^{\frac{1}{m}} \right )^{n} = \left ( a^{n} \right )^{\frac{1}{m}} \end{split}$

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$\begin{split} \sqrt[n]{a} &= a^{\frac{1}{n}} \\ \\ \sqrt[n]{ab} &= \sqrt[n]{a} \sqrt[n]{b} \\ \\ \sqrt[m]{\sqrt[n]{a}} &= \sqrt[nm]{a} \\ \\ \sqrt[n]{\frac{a}{b}} &= \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \\ \\ \sqrt[n]{a^{n}} &= a \text{, if }n \text{ is odd} \\ \\ \sqrt[n]{a^{n}} &= \left | a \right | \text{, if }n \text{ is even} \end{split}$

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Properties of Inequalities

$\begin{split} \text{If } a < b \text{ then } a+c < b+c \text{ and } a-c < b-c \\ \\ \text{If } a < b \text{ and } c > 0 \text{ then } ac < bc \text{ and } \frac{a}{c} < \frac{b}{c} \\ \\ \text{If } a < b \text{ and } c < 0 \text{ then } ac > bc \text{ and } \frac{a}{c} > \frac{b}{c} \end{split}$

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Properties of Absolute Value

$\begin{split} |a| &= \left\{\begin{matrix} a \text{, if } a \geq 0\\ -a \text{, if } a < 0 \end{matrix}\right. \\ \\ \left | a \right | &\geq 0 \\ \\ \left | -a \right | &= \left | a \right | \\ \\ \left | ab \right | &= \left | a \right | \left | b \right | \\ \\ \left | \frac{a}{b} \right | &= \frac{\left | a \right |}{\left | b \right |} \\ \\ \left | a+b \right | &\leq \left | a \right | + \left | b \right | \text{ Triangle Inequality} \end{split}$

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Distance Formula

$\begin{split} \text{If } P_{1} = (x_{1}, y_{1}) \text{ and } P_{2} = (x_{2}, y_{2}) \text{ are two points, then the distance between them is:} \\ d \left ( P_{1}, P_{2} \right ) = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}} \end{split}$

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Complex Numbers

$\begin{split} i &= \sqrt{-1}\\ \\ i^{2} &= -1 \\ \\ \sqrt{-a} &= i \sqrt{a} \\ \\ (a+bi)+(c+di) &= a+c+(b+d)i \\ \\ (a+bi)-(c+di) &= a-c+(b-d)i \\ \\ (a+bi)(c+di) &= ac-bd+(ad+bc)i \\ \\ (a+bi)(a-bi) &= a^{2} + b^{2} \\ \\ |a+bi| &= \sqrt{a^{2}+b^{2}} \quad \text{ Complex Modulus}\\ \\ \overline{(a+bi)} &= a-bi \quad \text{ Complex Conjugate}\\ \\ \overline{(a+bi)} (a+bi) &= \left | a+bi \right |^{2} \end{split}$

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Logarithms and Log Properties

Definition
$y = \log_{b}{x} \text{ is equivalent to } x=b^{y}$

Example
$\log_{5}{125} = 3 \text{ because } 5^{3}=125$

Special Logarithms
$\begin{split} \ln{x} &= \log_{e}{x} \quad \text{natural logarithm, where } e = 2.718281828… \\ \\ \log{x} &= \log_{10}{x} \quad \text{common logarithm} \end{split}$

Logarithm Properties
$\begin{split} \log_{b}{b} &= 1 \\ \\ \log_{b}{1} &= 0 \\ \\ \log_{b}{b^{x}} &= x \\ \\ b^{\log_{b}{x}} &= x \\ \\ \log_{b} \left ( x^{r} \right ) &= r \log_{b}{x} \\ \\ \log_{b}{\left ( xy \right )} &= \log_{b}{x} + \log_{b}{y} \\ \\ \log_{b}{\left ( \frac{x}{y} \right )} &= \log_{b}{x} – \log_{b}{y} \\ \\ \text{The domain of } \log_{b}{x} \text{ is } x>0 \end{split}$

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Factoring Formulas

$\begin{split} x^{2}-a^{2} &= \left ( x+a\right ) \left ( x-a \right ) \\ \\ x^{2} + 2ax + a^{2} &= \left ( x+a \right ) ^{2} \\ \\ x^{2} – 2ax + a^{2} &= \left ( x-a \right ) ^{2} \\ \\ x^{2}+ \left ( a+b \right ) x + ab &= \left ( x+a \right ) \left ( x+b \right ) \\ \\ x^{3}+3ax^{2}+3a^{2}x+a^{3} &= \left ( x+a \right ) ^{3} \\ \\ x^{3}-3ax^{2}+3a^{2}x-a^{3} &= \left ( x-a \right ) ^{3} \\ \\ x^{3} + a^{3} &= \left ( x+a \right ) \left ( x^{2} – ax + a^{2} \right ) \\ \\ x^{3} – a^{3} &= \left ( x-a \right ) \left ( x^{2} + ax + a^{2} \right ) \\ \\ x^{2n} – a^{2n} &= \left ( x^{n} – a^{n} \right ) \left ( x^{n} + a^{n} \right ) \\ \\ \text{If }n \text{ is odd, then:} \\ \\ x^{n} – a^{n} &= \left ( x-a \right ) \left ( x^{n-1} + ax^{n-2} + … + a^{n-1} \right ) \\ \\ x^{n} + a^{n} &= \left ( x+a \right ) \left ( x^{n-1} – ax^{n-2} + a^{2}x^{n-3} … + a^{n-1} \right ) \end{split}$

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$\begin{split} \text{Solve } ax^{2}+bx+c &= 0 \text{, } a \neq 0 \\ \\ x &= \frac{-b \pm \sqrt{\Delta}}{2a} \\ \\ \text{where } \Delta &= b^{2}-4ac \\ \\ \text{If } \Delta & < 0 \text{ the equation has two real unequal solutions} \\ \\ \text{If } \Delta & = 0 \text{ the equation has one real repeated solution} \\ \\ \text{If } \Delta & < 0 \text{ the equation has two complex solutions} \quad \end{split}$

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Square Root Property

$\text{If } x^{2} = p \text{ then } x = \pm \sqrt{p}$

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Absolute Value Equations/Inequalities

$\begin{split} \text{If } b \text{ is a positive number} \\ \\ |p| = b \quad & \Rightarrow \quad p = -b \text{ or } p = b \\ \\ |p| < b \quad & \Rightarrow \quad -b < p < b \\ \\ |p| > b \quad & \Rightarrow \quad p < -b \text{ or } p > b \end{split}$

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Completing the Square

$\text{Solve } 2x^{2}-6x-10=0$

1. Divide by the coefficient of x2 (which is 2 in this example):
$x^{2} – 3x – 5 = 0$

2. Move the constant to the other side:
$x^{2} – 3x = 5$

3. Take half the coefficient of x, square it and add it to both sides:
$x^{2} – 3x + \left ( – \frac{3}{2} \right )^{2} = 5 + \left ( – \frac{3}{2} \right )^{2} = 5 + \frac{9}{4} = \frac{29}{4}$

4. Factor the left side:
$\left ( x – \frac{3}{2} \right )^{2}= \frac{29}{4}$

5. Use Square Root Property:
$x – \frac{3}{2} = \pm \sqrt{\frac{29}{4}} = \pm \frac{\sqrt{29}}{2}$

6. Solve for x:
$x = \frac{3}{2} \pm \frac{\sqrt{29}}{2}$

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Constant Function

$y=a \quad \text{or} \quad f(x)=a$
Graph is a horizontal line passing through the point (0, a).

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Line/Linear Function

$y=mx+b \quad \text{or} \quad f(x)=mx+b$
Graph is a line with point (0, b) and slope m.

Slope
Slope of the line containing the two points (x1, y1) and (x2, y2) is:
$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{\text{rise}}{\text{run}}$

Slope – intercept form
The equation of the line with slope m and y-intercept (0, b) is:
$y=mx+b$

Point – slope form
The equation of the line with slope m and and passing through the point (x1, y1) is:
$y=y_{1} + m(x-x_{1})$

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Vertex form
$\begin{split} y &= a \left ( x-h \right )^{2} + k\\ f(x) &= a \left ( x-h \right )^{2} + k \end{split}$

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h, k).

General form
$\begin{split} y &= ax^{2}+bx+c \\ f(x) &= ax^{2}+bx+c \end{split}$

The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at:
$\left ( – \frac{b}{2a}, f \left ( – \frac{b}{2a} \right ) \right )$

$\begin{split} x &= ay^{2}+by+c \\ g(y) &= ay^{2}+by+c \end{split}$

The graph is a parabola that opens right if a > 0 or down if a < 0 and has a vertex at:
$\left ( g \left ( – \frac{b}{2a} \right ), – \frac{b}{2a} \right )$

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Circle

$\left ( x-h \right )^{2} + \left ( y-k \right )^{2} = r^{2}$

Graph is a circle with radius r and centre (h, k).

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Ellipse

$\frac{\left ( x-h \right )^{2}}{a^{2}} + \frac{\left ( y-h \right )^{2}}{b^{2}} = 1$

Graph is an ellipse with centre (h, k) with vertices a units right/left from the centre and vertices b units up/down from the centre.

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Hyperbola

Left-right vertices
$\frac{\left ( x-h \right )^{2}}{a^{2}} – \frac{\left ( y-h \right )^{2}}{b^{2}} = 1$

Graph is a hyperbola that opens left and right, has a centre at (h, k), vertices a units left/right of centre and asymptotes that pass through centre with slope:
$\pm \frac{b}{a}$

Up-down vertices
$\frac{\left ( y-k \right )^{2}}{b^{2}} – \frac{\left ( x-h \right )^{2}}{a^{2}} = 1$

Graph is a hyperbola that opens up and down, has a centre at (h, k), vertices b units up/down from the centre and asymptotes that pass through centre with slope:
$\pm \frac{b}{a}$

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Error and Reason/Correct/Justification/Example

$\begin{matrix} \mathbf{Error} & \quad \mathbf{Reason/Correct/Justification/Example}\\ \\ \frac{2}{0} \neq 0 \text{ and } \frac{2}{0} \neq 2 & \text{Division by zero is undefined!} \\ \\ -3^{2} \ne 9 & -3^{2}=-9 \text{, } (-3)^{2}=9 \text{ Watch parenthesis!} \\ \\ \left ( x^{2} \right )^{3} \ne x^{5} & \left ( x^{2} \right )^{3} = x^{2}x^{2}x^{2} = x^{6} \\ \\ \frac{a}{b+c} \ne \frac{a}{b} + \frac{a}{c} & \frac{1}{2} = \frac{1}{1+1} \ne \frac{1}{1}+\frac{1}{1} = 2 \\ \\ \frac{1}{x^{2}+x^3} \ne x^{-2} + x^{-3} & \text{A more complex version of the previous error} \\ \\ \frac{\not{a}+bx}{\not{a}} \ne 1+bx & \frac{a+bx}{a}=\frac{a}{a}+\frac{bx}{a}=1+\frac{bx}{a} \\ \quad & \text{Beware of incorrect canceling!} \\ \\ -a(x-1) \ne -ax-a & -a(x-1) = -ax+a \\ \quad & \text{Make sure you distribute the ”-”!} \\ \\ \left ( x+a \right )^{2} \ne x^{2}+a^{2} & \left ( x+a \right )^{2}=\left ( x+a \right )\left ( x+a \right )=x^{2}+2ax+x^{2} \\ \\ \sqrt{x^{2}+a^{2}} \ne x+a & 5 = \sqrt{25} = \sqrt{3^{2}+4^{2}} \ne \sqrt{3^{2}} + \sqrt{4^{2}} = 3+4 =7 \\ \\ \sqrt{x+a} \ne \sqrt{x} + \sqrt{a} & \text{See previous error.} \\ \\ \left ( x+a \right )^{n} \ne x^{n}+a^{n} & \text{More general versions of previous three errors.} \\ \sqrt[n]{x+a} \ne \sqrt[n]{x} + \sqrt[n]{a} & \quad \\ \\ 2 \left ( x+1 \right )^{2} \ne \left ( 2x+2 \right )^{2} & 2 \left ( x+1 \right )^{2} = 2 \left ( x^{2}+2x+1 \right ) = 2x^{2} + 4x + 2 \\ \quad & \left ( 2x+2 \right )^{2} = 4x^{2}+8x+4 \\ \quad & \text{Square first then distribute!} \\ \\ \left ( 2x+2 \right )^{2} \ne 2 \left ( x+1 \right )^{2} & \text{See previous example.} \\ \quad & \text{You can not factor out a constant,} \\ \quad & \text{if there is a power on the parenthesis!} \\ \\ \sqrt{-x^{2}+a^{2}} \ne – \sqrt{x^{2}+a^{2}} & \sqrt{-x^{2}+a^{2}} = \left ( -x^{2}+a^{2} \right )^{\frac{1}{2}} \\ \quad & \text{Now see the previous error.} \\ \\ \frac{a}{\left ( \frac{b}{c} \right ) } \ne \frac{ab}{c} & \frac{a}{\left ( \frac{b}{c} \right )} = \frac{\left ( \frac{a}{1} \right )}{\left ( \frac{b}{c} \right )} = \left ( \frac{a}{1} \right ) \left ( \frac{c}{b} \right ) = \frac{ac}{b}\\ \\ \frac{\left ( \frac{a}{b} \right )}{c} \ne \frac{ac}{b} & \frac{\left ( \frac{a}{b} \right )}{c} = \frac{\left ( \frac{a}{b} \right )}{\left ( \frac{c}{1} \right )} = \left ( \frac{a}{b} \right ) \left ( \frac{1}{c} \right ) = \frac{a}{bc} \end{matrix}$

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Reference:
https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx