Basic Properties & Facts
- Arithmetic Operations
- Exponent Properties
- Properties of Radicals
- Properties of Inequalities
- Properties of Absolute Value
- Distance Formula
- Complex Numbers
- Logarithms and Log Properties
Factoring and Solving
- Factoring Formulas
- Quadratic Formula
- Square Root Property
- Absolute Value Equations/Inequalities
- Completing the Square
Functions and Graphs
Common Algebraic Errors
Arithmetic Operations
\[ \begin{split}ab+ac &= a \left(b+c \right)\\
\\
\frac{\left( \frac{a}{b} \right)}{c} &= \frac{a}{bc}\\
\\
\frac{a}{b} + \frac{c}{d} &= \frac{ad+bc}{bd}\\
\\
\frac{a-b}{c-d} &= \frac{b-a}{d-c}\\
\\
\frac{ab+ac}{a} &= b+c \text{, } a \neq 0\\
\\
a \left(\frac{b}{c} \right) &= \frac{ab}{c}\\
\\
\frac{a}{\left ( \frac{b}{c} \right )} &= \frac{ac}{b}\\
\\
\frac{a}{b} – \frac{c}{d} &= \frac{ad – bc}{bd}\\
\\
\frac{a+b}{c} &= \frac{a}{c} + \frac{b}{c}\\
\\
\frac{\left( \frac{a}{b} \right)}{\left( \frac{c}{d} \right)} &= \frac{ad}{bc}
\end{split} \]
Exponent Properties
\[ \begin{split}a^{n}a^{m} &= a^{m+n} \\
\\
\left ( a^{n} \right )^{m} &= a^{nm} \\
\\
\left ( ab \right )^{n} &= a^{n}b^{n} \\
\\
a^{-n} &= \frac{1}{a^{n}} \\
\\
\left ( \frac{a}{b} \right )^{-n} &= \left ( \frac{b}{a} \right )^{n} = \frac{b^{n}}{a^{n}} \\
\\
\frac{a^{n}}{a^{m}} &= a^{n-m} = \frac{1}{a^{m-n}} \\
\\
a^{0} &= 1 \text{, } a \neq 0 \\
\\
\left ( \frac{a}{b} \right )^{n} &= \frac{a^{n}}{b^{n}} \\
\\
\frac{1}{a^{-n}} &= a^{n}\\
\\
a^{\frac{n}{m}} &= \left ( a^{\frac{1}{m}} \right )^{n} = \left ( a^{n} \right )^{\frac{1}{m}}
\end{split} \]
Properties of Radicals
\[ \begin{split}\sqrt[n]{a} &= a^{\frac{1}{n}} \\
\\
\sqrt[n]{ab} &= \sqrt[n]{a} \sqrt[n]{b} \\
\\
\sqrt[m]{\sqrt[n]{a}} &= \sqrt[nm]{a} \\
\\
\sqrt[n]{\frac{a}{b}} &= \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \\
\\
\sqrt[n]{a^{n}} &= a \text{, if }n \text{ is odd} \\
\\
\sqrt[n]{a^{n}} &= \left | a \right | \text{, if }n \text{ is even}
\end{split} \]
Properties of Inequalities
\[ \begin{split}\text{If } a < b \text{ then } a+c < b+c \text{ and } a-c < b-c \\
\\
\text{If } a < b \text{ and } c > 0 \text{ then } ac < bc \text{ and } \frac{a}{c} < \frac{b}{c} \\
\\
\text{If } a < b \text{ and } c < 0 \text{ then } ac > bc \text{ and } \frac{a}{c} > \frac{b}{c}
\end{split} \]
Properties of Absolute Value
\[ \begin{split}|a| &= \left\{\begin{matrix}
a \text{, if } a \geq 0\\
-a \text{, if } a < 0
\end{matrix}\right. \\
\\
\left | a \right | &\geq 0 \\
\\
\left | -a \right | &= \left | a \right | \\
\\
\left | ab \right | &= \left | a \right | \left | b \right | \\
\\
\left | \frac{a}{b} \right | &= \frac{\left | a \right |}{\left | b \right |} \\
\\
\left | a+b \right | &\leq \left | a \right | + \left | b \right | \text{ Triangle Inequality}
\end{split} \]
Distance Formula
\[ \begin{split}\text{If } P_{1} = (x_{1}, y_{1}) \text{ and } P_{2} = (x_{2}, y_{2}) \text{ are two points, then the distance between them is:} \\
d \left ( P_{1}, P_{2} \right ) = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}
\end{split} \]
Complex Numbers
\[ \begin{split}i &= \sqrt{-1}\\
\\
i^{2} &= -1 \\
\\
\sqrt{-a} &= i \sqrt{a} \\
\\
(a+bi)+(c+di) &= a+c+(b+d)i \\
\\
(a+bi)-(c+di) &= a-c+(b-d)i \\
\\
(a+bi)(c+di) &= ac-bd+(ad+bc)i \\
\\
(a+bi)(a-bi) &= a^{2} + b^{2} \\
\\
|a+bi| &= \sqrt{a^{2}+b^{2}} \quad \text{ Complex Modulus}\\
\\
\overline{(a+bi)} &= a-bi \quad \text{ Complex Conjugate}\\
\\
\overline{(a+bi)} (a+bi) &= \left | a+bi \right |^{2}
\end{split} \]
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
\[ \begin{split}
\log_{b}{b} &= 1 \\
\\
\log_{b}{1} &= 0 \\
\\
\log_{b}{b^{x}} &= x \\
\\
b^{\log_{b}{x}} &= x \\
\\
\log_{b} \left ( x^{r} \right ) &= r \log_{b}{x} \\
\\
\log_{b}{\left ( xy \right )} &= \log_{b}{x} + \log_{b}{y} \\
\\
\log_{b}{\left ( \frac{x}{y} \right )} &= \log_{b}{x} – \log_{b}{y} \\
\\
\text{The domain of } \log_{b}{x} \text{ is } x>0
\end{split} \]
Factoring Formulas
\[ \begin{split}x^{2}-a^{2} &= \left ( x+a\right ) \left ( x-a \right ) \\
\\
x^{2} + 2ax + a^{2} &= \left ( x+a \right ) ^{2} \\
\\
x^{2} – 2ax + a^{2} &= \left ( x-a \right ) ^{2} \\
\\
x^{2}+ \left ( a+b \right ) x + ab &= \left ( x+a \right ) \left ( x+b \right ) \\
\\
x^{3}+3ax^{2}+3a^{2}x+a^{3} &= \left ( x+a \right ) ^{3} \\
\\
x^{3}-3ax^{2}+3a^{2}x-a^{3} &= \left ( x-a \right ) ^{3} \\
\\
x^{3} + a^{3} &= \left ( x+a \right ) \left ( x^{2} – ax + a^{2} \right ) \\
\\
x^{3} – a^{3} &= \left ( x-a \right ) \left ( x^{2} + ax + a^{2} \right ) \\
\\
x^{2n} – a^{2n} &= \left ( x^{n} – a^{n} \right ) \left ( x^{n} + a^{n} \right ) \\
\\
\text{If }n \text{ is odd, then:} \\
\\
x^{n} – a^{n} &= \left ( x-a \right ) \left ( x^{n-1} + ax^{n-2} + … + a^{n-1} \right ) \\
\\
x^{n} + a^{n} &= \left ( x+a \right ) \left ( x^{n-1} – ax^{n-2} + a^{2}x^{n-3} … + a^{n-1} \right )
\end{split} \]
Quadratic Formula
\[ \begin{split}\text{Solve } ax^{2}+bx+c &= 0 \text{, } a \neq 0 \\
\\
x &= \frac{-b \pm \sqrt{\Delta}}{2a} \\
\\
\text{where } \Delta &= b^{2}-4ac \\
\\
\text{If } \Delta & < 0 \text{ the equation has two real unequal solutions} \\
\\
\text{If } \Delta & = 0 \text{ the equation has one real repeated solution} \\
\\
\text{If } \Delta & < 0 \text{ the equation has two complex solutions} \quad
\end{split} \]
Square Root Property
\[ \text{If } x^{2} = p \text{ then } x = \pm \sqrt{p} \]Absolute Value Equations/Inequalities
\[ \begin{split}\text{If } b \text{ is a positive number} \\
\\
|p| = b \quad & \Rightarrow \quad p = -b \text{ or } p = b \\
\\
|p| < b \quad & \Rightarrow \quad -b < p < b \\
\\
|p| > b \quad & \Rightarrow \quad p < -b \text{ or } p > b
\end{split} \]
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:
\[x = \frac{3}{2} \pm \frac{\sqrt{29}}{2}\]
Constant Function
\[ y=a \quad \text{or} \quad f(x)=a\]
Graph is a horizontal line passing through the point (0, a).
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:
\[y=y_{1} + m(x-x_{1})\]
Parabola/Quadratic Function
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:
\[\left ( – \frac{b}{2a}, f \left ( – \frac{b}{2a} \right ) \right )\]
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:
\[\left ( g \left ( – \frac{b}{2a} \right ), – \frac{b}{2a} \right )\]
Circle
\[\left ( x-h \right )^{2} + \left ( y-k \right )^{2} = r^{2}\]Graph is a circle with radius r and centre (h, k).
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.
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:
\[\pm \frac{b}{a}\]
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}\]
Reference:
https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx