## CLEP College Algebra Formulas

Preparing for the CLEP College Algebra test with a few weeks or even a few days to study? Think about it this way you get your dose of stress, which is good for you.

When you need to remember a math formula or explanation, this list of formulas and explanations is perfect. Reviewing them all before test day will help you learn the necessary formulas that can be applied in further math topics. You'll have an easier time memorizing concepts if they are used appropriately!

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*Place Value*

*Place Value*

The value of the place, or position, of a digit in a number.

Example: In 456, the 5 is in “tens” position.

*Fractions*

*Fractions*

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:

\(\frac{a}{b} × \frac{c}{d}=\frac{a×c}{b×d}\)

\(\frac{a}{b} ÷ \frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\)

*Comparing Numbers Signs*

*Comparing Numbers Signs*

Equal to \(=\)

Less than \( <\)

Greater than \(>\)

Greater than or equal \(≥\)

Less than or equal \(≤\)

*Rounding*

*Rounding*

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.

*Whole Number*

*Whole Number*

The numbers \( \{0,1,2,3,…\} \)

*Estimates*

*Estimates*

Find a number close to the exact answer.

*Decimals*

*Decimals*

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

*Mixed Numbers*

*Mixed Numbers*

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

*Factoring Numbers*

*Factoring Numbers*

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 Rules*

*Divisibility Rules*

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

*Greatest Common Factor*

*Greatest Common Factor*

Multiply common prime factors

Example:\( 200=2×2×2×5×5 60=2×2×3×5\)

GCF \((200,60)=2×2×5=20\)

*Least Common Multiple*

*Least Common Multiple*

Check multiples of the largest number

Example: LCM (200, 60): 200 (no), 400 (no), 600 (yes!)

*Integers*

*Integers*

\( \{…,-3,-2,-1,0,1,2,3,…\} \)

Includes: zero, counting numbers, and the negative of the counting numbers

*Real Numbers*

*Real Numbers*

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

*Order of Operations*

*Order of Operations*

PEMDAS

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

*Absolute Value*

*Absolute Value*

Refers to the distance of a number from , the distances are positive as absolute value of a number cannot be negative. \(|-22|=22\)

or \(|x| =\begin{cases}x \ for \ x≥0 \\x \ for \ x < 0\end{cases} \)

\(|x|<n⇒-n<x<n\)

\(|x|>n⇒x<-n \ or \ x>n\)

*Ratios*

*Ratios*

A ratio is a comparison of two numbers by division.

Example: \(3 : 5\), or \(\frac{3}{5}\)

*Percentages*

*Percentages*

Use the following formula to find part, whole, or percent

part \(=\frac{percent}{100}×whole\)

*Proportional Ratios*

*Proportional Ratios*

A proportion means that two ratios are equal. It can be written in two ways:

\(\frac{a}{b}=\frac{c}{d}\) , \(a: b = c: d \)

*Percent of Change*

*Percent of Change*

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

*Markup*

*Markup*

Markup \(=\) selling price \(–\) cost

Markup rate \(=\) markup divided by the cost

*Discount*

*Discount*

Multiply the regular price by the rate of discount

Selling price \(=\) original price \(–\) discount

*Expressions and Variables*

*Expressions and Variables*

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

*Tax*

*Tax*

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

*Distributive Property*

*Distributive Property*

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

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*Polynomial*

*Polynomial*

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

*Equations*

*Equations*

The values of two mathematical expressions are equal.

\(ax+b=c\)

*Systems of Equations*

*Systems of Equations*

Two or more equations working together.

example: \( \begin{cases}-2x \ + \ 2y \ =4\\-2x \ + \ y \ = \ 3\end{cases} \)

*Solving Systems of Equations by **Substitution*

*Solving Systems of Equations by*

*Substitution*

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 \(x \ = \ 1 \ + \ y\) \(⇒ \ x \ = \ 1 \ + \ 2 \ = \ 3\)

*Solving Systems of Equations by **Elimination*

*Solving Systems of Equations by*

*Elimination*

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

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

*Functions*

*Functions*

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.

*Inequalities*

*Inequalities*

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

*Lines (Linear Functions)*

*Lines (Linear Functions)*

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

*Distance from A to B:*

*Distance from A to B:*

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

*Parallel and Perpendicular lines:*

*Parallel and Perpendicular lines:*

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)

*Mid-point of the segment AB:*

*Mid-point of the segment AB:*

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

*Slope of the line:*

*Slope of the line:*

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

*Point-slope form:*

*Point-slope form:*

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

\((y-y_{1})=m \ (x-x_{1})\).

*Intersecting lines:*

*Intersecting lines:*

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

*Slope-intercept form:*

*Slope-intercept form:*

given the slope m and the y-intercept b, then the equation of the line is:

\(y=mx+b\).

**Transversal**: *Parallel lines:*

*Parallel lines:*

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.

*Parabolas:*

*Parabolas:*

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: \(x=-\frac{b}{2a}\).

*Factoring:*

*Factoring:*

“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 \(4x-(3-x)=2 ⇒ 5x-3=2\) \(⇒ x=1,y=2\).

*Exponents:*

*Exponents:*

Refers to the number of times a number is multiplied by itself.

\(8 = 2 × 2 × 2 = 2^3\)

*Scientific Notation:*

*Scientific Notation:*

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: \(m \times 10^n\)

Decimal notation |
Scientific notation |

5 | \(5×10^0\) |

\(-25,000\) | \(-2.5×10^4\) |

0.5 | \(5×10^{-1}\) |

2,122.456 | \(2,122456×10^3\) |

*Square:*

*Square:*

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

*Square Roots:*

*Square Roots:*

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

\(r\) is a square root of \(x\)

*Pythagorean Theorem:*

*Pythagorean Theorem:*

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

##### Triangles

*Right triangles:*

*Right triangles:*

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.

**All triangles:**

**All triangles:**

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:

*Equilateral:*

*Equilateral:*

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

*Isosceles:*

*Isosceles:*

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

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*Similar:*

*Similar:*

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 \(1\).

*Circles*

*Circles*

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): \((x \ - \ h)^2 \ + \ (y \ - \ k)^2 \ = \ r^2\).

**Rectangles**

**Rectangles**

(Square if *l=w*)

Area=*lw*

**Parallelogram**

**Parallelogram**

(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

\((n-2) .180^\circ\).

*Area of a parallelogram:*

*Area of a parallelogram:*

\(A = bh\)

*Area of a trapezoid:*

*Area of a trapezoid:*

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

*Surface Area and Volume of a rectangular/right prism:*

*Surface Area and Volume of a rectangular/right prism:*

\(SA=ph+2B\)

\(V=Bh\)

**Surface Area and Volume of a cylinder:**

**Surface Area and Volume of a cylinder:**

\(SA =2πrh+2πr^2\)

\(V =πr^2 h \)

**Surface Area and Volume of a Pyramid**

**Surface Area and Volume of a Pyramid**

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

\(V=\frac{1}{3}\ bh\)

**Surface Area and Volume of a Cone**

**Surface Area and Volume of a Cone**

\(SA =πrs+πr^2\)

\(V=\frac{1}{3} \ πr^2 \ h\)

**Surface Area and Volume of a Sphere**

**Surface Area and Volume of a Sphere**

\(SA =4πr^2\)

\(V =\frac{4}{3} \ πr^3\)

(p \(=\) perimeter of base B; \(π ~ 3.14 \))

*Solids*

*Solids*

Rectangular Solid

Volume \(= \ l \times w \times h\)

Area \(= \ 2(lw \ + \ wh \ + \ lh)\)

Right Cylinder

Volume \(= \ πr^2 \ h\)

Area \(= \ 2πr \ (r \ + \ h)\)

**Quadratic formula**:

**Quadratic formula**:

\( x=\frac{-b±\sqrt{b^2-4ac}}{2a}\)

**Simple interest**:

**Simple interest**

\(I=prt\)

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

**mean**:

**mean**

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

**mode:**

**mode:**

value in the list that appears most often

**range:**

**range:**

largest value \(-\) smallest value

*Median*

*Median*

Middle value in the list (which must be sorted)

Example: median of

\( \{3,10,9,27,50\} = 10\)

Example: median of

\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)

*Sum*

*Sum*

*average *\(×\) (*number of terms*)

*Average*

*Average*

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

*Average speed*

*Average speed*

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

*Probability*

*Probability*

\(\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}\)

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).

*Powers, Exponents, Roots*

*Powers, Exponents, Roots*

\(x^a .x^b=x^{a+b}\)

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

\(\frac{1}{x^b }= x^{-b}\)

\((x^a)^b=x^{a.b}\)

\((xy)^a= x^a .y^a\)

\(x^0=1\)

\(\sqrt{xy}=\sqrt{x} .\sqrt{y}\)

\((-1)^n=-1\), if n is odd.

\((-1)^n=+1\), if n is even.

If \(0<x<1\), then

\(0<x^3<x^2<x<\sqrt{x}<\sqrt{3x}<1\).

##### Interest

*Simple Interest*

*Simple Interest*

The charge for borrowing money or the return for lending it.

Interest = principal \(×\) rate \(×\) time

OR

\(I=prt\)

*Compound Interest*

*Compound Interest*

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

*Arithmetic Sequences*

*Arithmetic Sequences*

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: \(S_n \ = \ (\frac{n}{2}) \times (t_1 \ + \ t_n)\)

*Geometric Sequences*

*Geometric Sequences*

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: \(S_n \ = \ t_1 \times \frac{(1 \ - \ r^n)}{(1 \ - \ r)}\)

##### Powers / Exponents

*Positive Exponents*

*Positive Exponents*

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, \((x)(x)(x)\).

*Negative Exponents*

*Negative Exponents*

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

\(x^{-3}=\frac{1}{x^3}\)

**Factorials**

**Factorials**

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 \(=2!=2× 1=2\)

**Multiplying Two Powers of the SAME Base**

**Multiplying Two Powers of the SAME Base**

When the bases are the same, you find the new power by just adding the exponents

\(x^a .x^b=x^{a+b }\)

*Multiplying Two Powers of Different Bases Same Exponent*

*Multiplying Two Powers of Different Bases Same Exponent*

If the bases are different but the exponents are the same, then you can combine them

\(x^a \times y^a \ = \ (xy)^a\)

*Powers of Powers*

*Powers of Powers*

For power of a power: you multiply the exponents.

\((x^a)^b=x^{(ab)}\)

*Dividing Powers*

*Dividing Powers*

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

*The Zero Exponent*

*The Zero Exponent*

Anything to the 0 power is 1.

\(x^0= 1\)

*Permutation:*

*Permutation:*

When different orderings of the same items are counted separately, we have a permutation problem:

\(_{n}P_{r}=\frac{n!}{(n-1)!}\)

*Combination:*

*Combination:*

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:

\(_{n}C_{r}=\frac{n!}{r!(n-1)!}\)

*Adding and Subtracting Matrices*

*Adding and Subtracting Matrices*

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. \(a_{ij} \ + \ b_{ij} \ + \ c_{ij} \ = \ d_{ij}\)

\( \begin{bmatrix}9 & -3\\\ 4 & 1\end{bmatrix} + \begin{bmatrix}5 & 2\\\ -1 & 6\end{bmatrix} + \begin{bmatrix}1 & 3\\\ 3 & -2\end{bmatrix} = \begin{bmatrix}15 & 2\\\ 6 & 5\end{bmatrix}\)

*Matrices Scalar Multiplication*

*Matrices Scalar Multiplication*

**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** \(3A \ = \begin{bmatrix}12 & 9\\\ 6 & 3\end{bmatrix}\)

*Laws of Logarithm*

*Laws of Logarithm*

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,** \(log_a \ \sqrt{N} \ = \ \frac{1}{2} \ log_a \ N\))

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*The Six Trig Ratios*

*The Six Trig Ratios*

Values for the common angles

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

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

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

*Trig Functions relationships*

*Trig Functions relationships*

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

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

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

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