CLEP College Algebra Formulas

The value of the place, or position, of a digit in a number.
Example: In 456, the 5 is in “tens” position. 

A number expressed in the form \(\frac{a}{b}\)
Adding and Subtracting with the same denominator:
\(\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}\)
\(\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}\)
Adding and Subtracting with the different denominator:
\(\frac{a}{b}+\frac{c}{d}=\frac{ad+cb}{bd}\)
\(\frac{a}{b}-\frac{c}{d}=\frac{ad-cb}{bd}\)
Multiplying and Dividing Fractions:
\(\frac{a}{b} × \frac{c}{d}=\frac{a×c}{b×d}\)
\(\frac{a}{b} ÷ \frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\)    

Equal to \(=\)
Less than \( <\)
Greater than \(>\)
Greater than or equal \(≥\)
Less than or equal \(≤\)  

Putting a number up or down to the nearest whole number or the nearest hundred, etc. 
Example: 64 rounded to the nearest ten is 60 , because 64 is closer to 60 than to 70.  

The numbers \( \{0,1,2,3,…\} \)

Find a number close to the exact answer.  

Is a fraction written in a special form. For example, instead of writing  \(\frac{1}{2}\) you can write \(0.5\).  

A number composed of a whole number and fraction. Example: \(2 \frac{2}{ 3}\) Converting between improper fractions and mixed numbers: \(a \frac{c}{b}=a+\frac{c}{b}= \frac{ab+ c}{b}\)

Factor a number means to break it up into numbers that can be multiplied together to get the original number. Example:\(12=2×2×3\)

Divisibility means that you are able to divide a number evenly. Example: 24 is divisible by 6, because \(24÷6=4\)

Multiply common prime factors
Example:\( 200=2×2×2×5×5 60=2×2×3×5\)
GCF \((200,60)=2×2×5=20\)

Check multiples of the largest number
Example: LCM (200, 60): 200 (no),  400 (no), 600 (yes!)

\( \{…,-3,-2,-1,0,1,2,3,…\} \)
Includes: zero, counting numbers, and the negative of the counting numbers

All numbers that are on number line. Integers plus fractions, decimals, and irrationals  etc.) (\(\sqrt{2},\sqrt{3},π\), etc.)  

PEMDAS
(parentheses / exponents / multiply / divide / add / subtract)  

Refers to the distance of a number from , the distances are positive as absolute value of a number cannot be negative. \(|-22|=22\)
or \(|x| =\begin{cases}x \ for \ x≥0 \\x \ for \ x < 0\end{cases} \)
\(|x|<n⇒-n<x<n\)
\(|x|>n⇒x<-n \ or \ x>n\)  

A ratio is a comparison of two numbers by division.
Example: \(3 : 5\), or \(\frac{3}{5}\)  

Use the following formula to find part, whole, or percent
part \(=\frac{percent}{100}×whole\)

A proportion means that two ratios are equal. It can be written in two ways:  
\(\frac{a}{b}=\frac{c}{d}\) , \(a: b = c: d  \)

\(\frac{New \ Value \ – \ Old \ Value}{Old Value}×100\%\)

Markup \(=\) selling price \(–\) cost
Markup rate \(=\) markup divided by the cost

Multiply the regular price by the rate of discount
Selling price \(=\) original price \(–\) discount

A variable is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numbers:
Addition: \(2+a\)  : \(2\) plus a
Subtraction: \(y-3\)  : \(y\) minus \(3\)
Division: \(\frac{4}{x}\)  : 4 divided by x
Multiplication: \(5a\)  : \(5\) times a

To find tax, multiply the tax rate to the taxable amount (income, property value, etc.)

\(a(b+c)=ab+ac\)

\(P(x)=a_{0} x^n+ a_{1} x^{n-1}+\)⋯\(+a_{n-2} x^2+a_{n-1} x+an\)

The values of two mathematical expressions are equal.
\(ax+b=c\)

Two or more equations working together.
example: \( \begin{cases}-2x \ + \ 2y \ =4\\-2x \ + \ y \ = \ 3\end{cases} \)

Consider the system of equations: \( \begin{cases}x \ - \ y \ = \ 1\\-2x \ + \ y \ = \ 6\end{cases} \)
Substitute \(x \ = \ 1 \ - \ y\) in the second equation: \(-2(1 \ - \ y) \ + \ y \ = \ 5 \ ⇒ \ y \ = \ 2\)
Substitute \(y \ = \ 2\) in \(x \ = \ 1 \ + \ y\) \(⇒ \ x \ = \ 1 \ + \ 2 \ = \ 3\)

Example:
\(\cfrac{\begin{align} x+2y =6 \\ + \ \ -x+y=3 \end{align}}{}\)
\(\cfrac{ \begin{align} 3y=9 \\ y=3 \end{align} }{\begin{align} x+6=6 \\ ⇒ x=0 \end{align}} \)

A function is a rule to go from one number (x) to another number (y), usually written \(y=f(x)\).For any given value of x, there can only be one corresponding value y. If \(y=kx\) for some number k (example: \(f(x)= 0.5 x\)), then y is said to be directly proportional to x. If y\(=\frac{k}{x }\) (example: f(x \(=\frac{5}{x}\)), then y is said to be inversely proportional to x. The graph of \(y=f(x )+k\) is the translation of the graph of \(y=f(x)\) by \((h,k)\) units in the plane. For example, \(y=f(x+3)\) shifts the graph of \(f(x)\) by 3 units to the left.  

Says that two values are not equal
\(a≠b\)   a not equal to b
\(a<b\)   a less than b
\(a>b\)   a greater than b
\(a≥b\)   a greater than or equal b
\(a≤b\)   a less than or equal b  

Consider the line that goes through points \(A(x_{1},y_{1}) \) and \(B(x_{2},y_{2})\).  

\(\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 }\)

Have equal slopes. Perpendicular lines (i.e., those that make a \(90^° \) angle where they intersect) have negative reciprocal slopes: \(m_{1}\) .\(m_{2}=-1\).
Parallel Lines (l \(\parallel\) m)

M (\(\frac{x_{1}+x_{2}}{2} , \frac{y_{1}+y_{2}}{2}\))  

\(\frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{rise}{run}\)

Given the slope m and a point \((x_{1},y_{1})\) on the line, the equation of the line is
\((y-y_{1})=m \ (x-x_{1})\).  

Opposite angles are equal. Also, each pair of angles along the same line add to \(180^°\). In the figure above, \(a+b=180^°\).

given the slope m and the y-intercept b, then the equation of the line is:
\(y=mx+b\).

Eight angles are formed when a line crosses two parallel lines. The four big angles (a) are equal, and the four small angles (b) are equal.

A parabola parallel to the y-axis is given by \(y=ax^2+bx+c\).
If \(a>0\), the parabola opens up.
If \(a<0\), the parabola opens down. The y-intercept is c, and the x-coordinate of the vertex is: \(x=-\frac{b}{2a}\).

“FOIL”
\((x+a)(x+b)\)
\(=x^2+(b+a)x +ab\)

“Difference of Squares”
\(a^2-b^2= (a+b)(a-b)\)
\(a^2+2ab+b^2=(a+b)(a+b) \)
\(a^2-2ab+b^2=(a-b)(a-b)\)

“Reverse FOIL”
\(x^2+(b+a)x+ab=\) \((x+a)(x+b)\)

You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and b which add to the number in front of the x, and which multiply to give the constant. For example, to factor \(x^2+5x+6\), the numbers add to 5 and multiply to 6, i.e.:
\(a=2\) and \(b=3\), so that \(x^2+5x+6=(x+2)(x+3)\).

To solve a quadratic such as \(x^2+bx+c=0\), first factor the left side to get \((x+a)(x+b)=0\), then set each part in parentheses equal to zero.
For example, \(x^2+4x+3= (x+3)(x+1)=0\) so that \(x=-3\) or \(x=-1\).

To solve two linear equations in x and y: use the first equation to substitute for a variable in the second. E.g., suppose \(x+y=3\) and \(4x-y=2\). The first equation gives y=3-x, so the second equation becomes \(4x-(3-x)=2 ⇒ 5x-3=2\) \(⇒ x=1,y=2\).

Refers to the number of times a number is multiplied by itself.
\(8 = 2 × 2 × 2 = 2^3\)

It is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
In scientific notation all numbers are written in this form: \(m \times 10^n\)

The number we get after multiplying an integer (not a fraction) by itself. Example: \(2×2=4,2^2=4\)

A square root of \(x\) is a number r whose square is \(x : r^2=x\)
\(r\) is a square root of \(x\)

\(a^2+b^2=c^2\)

A good example of a right triangle is one with a=3, b=4, and c=5, also called a \( 3-4-5\) right triangle. Note that multiples of these numbers are also right triangles. For example, if you multiply these numbers by 2, you get a=6, b=8  and
\(c=10(6-8-10)\), which is also a right triangle.

Area \(=\frac{1}{2}\) \(b \times h\)
Angles on the inside of any triangle add up to \(180^\circ\).
The length of one side of any triangle is always less than the sum and more than the difference of the lengths of the other two sides.
An exterior angle of any triangle is equal to the sum of the two remote interior angles. Other important triangles:

These triangles have three equal sides, and all three angles are \(60^\circ\).  

An isosceles triangle has two equal sides. The “base” angles (the ones opposite the two sides) are equal (see the \(45^\circ\)  triangle above).  

Two or more triangles are similar if they have the same shape. The corresponding angles are equal, and the corresponding sides are in proportion. For example, the \(3-4-5\) triangle and the \(6-8-10\) triangle from before are similar since their sides are in a ratio of \(2\) to \(1\).

Area \(=πr^2\)
Circumference \(=2πr\)
Full circle \(=360^\circ\)

Length Of Arc \(= \ \frac{n°}{360°} \times 2πr\)
Area Of Sector \(= \ \frac{n°}{360°} \times πr^2\)
 
Equation of the circle (above left figure): \((x \ - \ h)^2 \ + \ (y \ - \ k)^2 \ = \ r^2\).

(Square if l=w)
Area=lw

(Rhombus if l=w)
Area=lh
Regular polygons are n-sided figures with all sides equal and all angles equal.
The sum of the inside angles of an n-sided regular polygon is
\((n-2) .180^\circ\).

 \(A = bh\)

\(A =\frac{1}{2} \ h (b_{1}+b_{2})\)

\(SA=ph+2B\)
\(V=Bh\)

\(SA =2πrh+2πr^2\)
\(V =πr^2 h  \)

\(SA=\frac{1}{2} \ ps+b\)
\(V=\frac{1}{3}\ bh\)

\(SA =πrs+πr^2\)
\(V=\frac{1}{3} \ πr^2 \ h\)

\(SA =4πr^2\)
\(V =\frac{4}{3} \ πr^3\)
(p \(=\) perimeter of base B; \(π ~ 3.14 \))

Rectangular Solid
Volume \(= \ l \times w \times h\)
Area \(= \ 2(lw \ + \ wh \ + \ lh)\)

Right Cylinder
Volume \(= \ πr^2 \ h\)
Area \(= \ 2πr \ (r \ + \ h)\)

\( x=\frac{-b±\sqrt{b^2-4ac}}{2a}\)

\(I=prt\)
(I = interest, p = principal, r = rate, t = time)

mean: \(\frac{sum \ of \ the \ data}{of \ data \ entires}\)

value in the list that appears most often

largest value \(-\) smallest value

Middle value in the list (which must be sorted)
Example: median of
\( \{3,10,9,27,50\} = 10\)
Example: median of
\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)

average \(×\) (number of terms)

\(   \frac{sum \ of \ terms}{number \ of \ terms}\)

\(\frac{total \ distance}{total \ time}\)

\(\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}\)
The probability of two different events A and B both happening is:
P(A and B)=p(A) .p(B)
as long as the events are independent (not mutually exclusive).

\(x^a .x^b=x^{a+b}\)
\(\frac{x^a}{x^b} = x^{a-b}\)
\(\frac{1}{x^b }= x^{-b}\)
\((x^a)^b=x^{a.b}\)
\((xy)^a= x^a .y^a\)
\(x^0=1\)
\(\sqrt{xy}=\sqrt{x} .\sqrt{y}\)
\((-1)^n=-1\), if n is odd.
\((-1)^n=+1\), if n is even.
If \(0<x<1\), then
\(0<x^3<x^2<x<\sqrt{x}<\sqrt{3x}<1\).

The charge for borrowing money or the return for lending it.
Interest = principal \(×\) rate \(×\) time
OR
\(I=prt\)

Interest computed on the accumulated unpaid interest as well as on the original principal.
A \(=P(1+r)^t\)
A= amount at end of time
P= principal (starting amount)
r= interest rate (change to a decimal i.e. \(50\%=0.50\))
t= number of years invested

Each term is equal to the previous term plus \(d\)
Sequence: \(t_1, \ t_1 \ + \ d, \ t_1 \ + \ 2d, \ . . .\)
The \(n\)th term is: \(t_n \ = \ t_1 \ + \ (n \ - \ 1)d\)
Number of integers from in to 
\(i_m \ = \ i_m \ - \ i_n \ + \ 1\)
Sum of \(n\) terms: \(S_n \ = \ (\frac{n}{2}) \times (t_1 \ + \ t_n)\)

each term is equal to the previous term times \(r\)
Sequence: \(t_1, \ t_1 \times r, \ t_1 \times r^2, \ ...\)
The \(n\)th term is \(t_n \ = \ t_1 \times r^{n \ - \ 1}\)
Sum of \(n\) terms: \(S_n \ = \ t_1 \times \frac{(1 \ - \ r^n)}{(1 \ - \ r)}\) 

An exponent is simply shorthand for multiplying that number of identical factors. So \(4^3\) is the same as (4)(4)(4), three identical factors of 4. And \(x^3\) is just three factors of x, \((x)(x)(x)\).

A negative exponent means to divide by that number of factors instead of multiplying.
So \(4^{-3}\) is the same as \( \frac{1}{4^3}\) and
\(x^{-3}=\frac{1}{x^3}\)

Factorial- the product of a number and all counting numbers below it.
8 factorial \(=8!=\)
\(8×7×6×5×4×3×2×1=40,320\)
5 factorial \(=5!=\)
\(5×4×3×2×1=120\)
2 factorial \(=2!=2× 1=2\)

When the bases are the same, you find the new power by just adding the exponents
\(x^a .x^b=x^{a+b }\)

If the bases are different but the exponents are the same, then you can combine them
\(x^a \times y^a \ = \ (xy)^a\)

For power of a power: you multiply the exponents.
\((x^a)^b=x^{(ab)}\)

\(\frac{x^a}{x^b} =x^a x^{-b}= x^{a-b}\)

Anything to the 0 power is 1.
\(x^0= 1\)

When different orderings of the same items are counted separately, we have a permutation problem:
\(_{n}P_{r}=\frac{n!}{(n-1)!}\)

The fundamental counting principle, as demonstrated above, is used any time the order of the outcomes is important.  When selecting objects from a group where the order is NOT important, we use the formula for COMBINATIONS:
The fundamental counting principle, as demonstrated above, is used any time the order of the outcomes is important. When selecting objects from a group where the order is NOT important, we use the formula for COMBINATIONS:
\(_{n}C_{r}=\frac{n!}{r!(n-1)!}\)

Matrices must have the same number of rows and columns, \(m \times n\)
Just add (subtract) the corresponding elements….. 
\(A \ + \ B \ + \ C \ = \ D\)    
i.e.  \(a_{ij} \ + \ b_{ij} \ + \ c_{ij} \ = \ d_{ij}\) 
\( \begin{bmatrix}9 & -3\\\ 4 & 1\end{bmatrix} + \begin{bmatrix}5 & 2\\\ -1 & 6\end{bmatrix} + \begin{bmatrix}1 & 3\\\ 3 & -2\end{bmatrix} = \begin{bmatrix}15 & 2\\\ 6 & 5\end{bmatrix}\)

If \(A \ = \begin{bmatrix}a_11 & a_12\\\ a_21 & a_22\end{bmatrix}\)

Then:  \(3A \ = \begin{bmatrix}3 \times a_11 & 3 \times a_12\\\ 3 \times a_21 & 3 \times a_22\end{bmatrix}\)

\(A \ = \begin{bmatrix}4 & 3\\\ 2 & 1\end{bmatrix}\) ,  then \(2A \ = \begin{bmatrix}8 & 6\\\ 4 & 2\end{bmatrix}\)
 
and \(3A \ = \begin{bmatrix}12 & 9\\\ 6 & 3\end{bmatrix}\)

1. \(log_a⁡ \ N \ = \ x \ ⟺ \ N \ = \ a^x\), where \(N \ > \ 0, \ a \ > \ 0, \ a \ ≠ \ 1\).
In particular, \(log_{10} ⁡N \ = \ x \ ⟺ \ N \ = \ 10^x\)

2. \(x \ = \ a^{log_a ⁡\ x}, \ log_a \ a^x \ = \ x\)

3. \(log_a⁡ \ 1 \ = \ 0, \ log_a⁡ \ a \ = \ 1\)

4. \(log_a⁡ \ MN \ = \ log_a⁡ \ M \ + \ log_a \ ⁡N\), where \(M, \ N \ > \ 0\)

5. \(log_a⁡ \ \frac{M}{N} \ = \ log_a⁡ \ M \ - \ log_a \ ⁡N\)

6.  \(log_a⁡ \ N^m \ = \ m \ log_a \ ⁡N\), where \(m\) is rational. (in particular, \(log_a \ ⁡\sqrt{N} \ = \ \frac{1}{2} \ log_a \ ⁡N\))

Values for the common angles

\(Sin(θ) \ = \ \frac{opp_.}{hip_.}\)         \(Csc(θ) \ = \ \frac{hip_.}{opp}\)

\(Cos(θ) \ = \ \frac{adj}{hip_.}\)         \(Sec(θ) \ = \ \frac{hip}{adj}\)

\(tan(θ) \ = \ \frac{opp_.}{adj_.}\)         \(tan(θ) \ = \ \frac{adj}{opp_.}\)

\(Tan(x) \ = \ \frac{Sin(x)}{Cos(x)}\)

\(Csc(x) \ = \ \frac{1}{Sin(x)}\)

\(Sec(x) \ = \ \frac{1}{Cos(x)}\)

\(Cot(x) \ = \ \frac{Cos(x)}{Sin(x)} \ = \ \frac{1}{Tan(x)}\)