AFOQT Arithmetic & Math - Formulas & Equations

Arc:

A portion of a circle's edge.

Algebraic Equations:

Add & Subtract (Ex: 5xy+7y+2yz+11xy-5yz=16xy+7y-3yz) Multiplication (First the coefficients then the variables) Monomial*Monomial (3x^(4)y^(2)z)(2y^(4)z(5))=6x^(4)y^(6)z(6)) Monomial*Polynomial ((2y^2)(y^(3)+2xy^(2)z+4z)=2y^(5)+4xy^(4)z+8y^(2)z) Binomial*Binomial ((5x+2)(3x+3) => FOIL = First, Outer, Inner, Last) Binomial*Polynomial ((x+3)(2x^(2)-5x-2); First term: x(2x^(2)-5x-2)=2x^(3)-5x^(2)-2x; Second term: 3(2x^(2)-5x-2)=6x^(2)-15x-6; Added together: 2x^(3)+x^(2)-17x-6.

Addition key words:

Added, combined, increased by, in all, total, perimeter, sum, & more than.

Supplementary Angles:

Adjacent angles that sum to 180 degrees.

Complementary Angles:

Adjacent angles that sum to 90 degrees.

Equilateral Triangle:

All angles are equal.

Vertical Angles:

Angles that are opposite of each other. They are always congruent (equal in measure).

Polygon:

Any enclosed plane shape with three or more connecting sides (ex. a triangle)

Linear Systems:

Substitution & Addition Methods to multiple equation linear systems: Substitution Ex: Solve: 3y-4+x=0 & 5x+6y=11 Step 1: Solve for one variable: 3y-4+x=0 3y+x=4 x=4-3y Step 2: Substitute in 2nd equation: 5(4-3y)+6y=11 20-15y+6y=11 20-9y=11 -9y+-9 y=1 Step 3: Substitute into 1st equation: 3(1)-4+x=0 -1+x=0 x=1 Solution: x=1, y=1 Addition Ex: Solve: 2x+4y=8 & 4x+2y=10 Step 1: Manipulate 1 equation to eliminate variable -2(2x+4y=8) -4x-8y=-16 (-4x-8y+-16)+(4x+2y=10) -6y=-6 y=1 Step 2: Plug into equation 2x+4(1)=8 2x+4=8 2x=4 x=2 Solution: x=2, y=1

Greatest Common Factor (GCF) is:

The greatest factor that divides two numbers. (Ex: 24 & 18; GCF = 6; greatest number that can divide both #'s)

Sum of Angles:

The sum of angles of a polygon can be calculated using (n-1)180 degrees, when n = the # of sides.

Isosceles Triangle:

Two sides and two angles are equal.

Distributive Property:

When a variable is placed outside of a parenthetical set, it is distributed to all the variables within that set. (Ex: 2x(3y+1)+6x = 6xy+2x+6x = 6xy +8x)

Exponent Rules:

a) x^0 = 1 b) x^1 = x c) x^a * x^b = x^(a+b) d) (xy)^a = (x^a)(y^a) e) (x^a)^b = x^a*b f) (x/y)^a = x^a / y^a g) x^a / y^b = x^a-b h) x^(1/a) = a*Sqrt(x) I) x^-a = 1/(x^a) j) (-x)^a = positive # if "a" is even; negative # if "a" is odd.

Differences 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)

How to find distance:

d=r*t (distance = rate x time)

Division key words:

distribute, share, average, per, out of, percent, and quotient.

Mode:

# that occurs most frequently in a set.

Positive & Negative Number Rules:

(+) + (-) = Subtract #'s; solution = larger # (-) + (-) = Negative # (-) * (-) = Positive # (-) * (+) = Negative # (-) / (-) = Positive # (-) / (+) = Negative #

Fundamental Counting Principle:

(The # of possibilities of an event happening) * (the # of possibilities of another event happening) = total # of possibilities. (Ex: Test choice of 5 questions with 4 answer choices for each question, how many test result possibilities are there? 4*4*4*4*4 = 1024 possible test results.)

Midpoint Equation:

(x1+x2 / 2) , (y1+y2 / 2)

Adding & subtracting fractions requires:

A common denominator. (Ex: 2/3 - 1/5 => 2/3*(5/5) - 1/5*(3/3) => 10/15 - 3/15 = 7/15)

Chord:

A line segment that connects two different points on a circle.

Transversal Line:

A line that crosses parallel lines.

Bisector:

Any line that cuts a line segment, angle, or polygon exactly in half.

Measurement of Angles:

Central Angle (c): 360 degrees/n Interior Angle (I): (n-2)180 degrees/n Exterior Angle (e): 260 degrees/n

To divide fractions:

Convert mixed fractions into single fractions, flip the second fraction, & multiply across.

To multiply fractions:

Convert mixed fractions to single fractions & multiply across. Reduce to lowest term if needed.

To subtract mixed fractions by:

Converting to single fractions.

A probability is found by:

Dividing the # desired outcomes by the # of possible outcomes. (Ex: Probability of picking a blue marble if 3 of the 15 are blue? 3/15 = 1/5; 1 in 5 chance of blue marble.)

Acute Triangle:

Each angle measures less than 90 degrees.

Geometric Sequence is:

Each term is equal to the previous term multiplied by x. (Ex: 2, 4, 8, 16; x=2)

Literal Equations are:

Equations with more than one variable. (Ex: Solve: 4x+3y=3x+2y; Combine like terms: 4x-3x=2y-3y; Solve for y: -y=x)

Prime Factorization:

Expand to prime number factors. (Ex: 104 = 2*2*2*13)

Quadratics:

Factoring: Converting ax^(2)+bx+c to factored form. Find two numbers that are factors of c and whose sum is b. Ex: 2x^(2)+12x+18=0 Step 1: Factor out common monomial 2(x^(2)-6x+9) Step 2: Find two #'s factors of 9 & = -6 w/ addition 2(x )(x ) -3 -3 Step 3: Fill binomials 2(x-3)(x-3) Step 4: To solve, set each to equal 0 x-3=0, So, x=3

To add mixed fractions:

First, work the whole # and then the fractions.

Regular Polygon:

Has all equal sides and equal angles (ex. square).

Subtraction key words:

How much more, less than, fewer than, exceeds, difference, & decreased.

Inequalities:

Inequalities are solved like linear & algebraic equations, except the sign must be reversed when dividing by a negative #. (Ex: -7x+2<6-5x; Combine like terms: -2x<4; Solve for x: x>-2) (Ex: -4<2x-2<6; Add 2 to each term to isolate x: -2<2x<8; Divide by 2: -1<x<4; Solution set is: [-1,4])

Equals key words:

Is, was, are, amounts to, & were.

The absolute value of a # is:

It's distance from zero, not its value. Absolute value equations will have two answers. (So in [x] = a, "x" will equal "-a" as well as "a.")

Perpendicular Lines:

Lines that intersect or cross to form 90 degree angles.

Parallel Lines:

Lines that will never intersect. Y II X means line Y is parallel to line X.

Mean:

Math term for "average". Total all terms and divide by the # of terms. (Ex: 24, 27, & 18; 24+27+18 = 69/3 = 23)

Acute Angle:

Measure of less than 90 degrees,

Obtuse Angle:

Measures greater than 90 degrees.

Median:

Middle # of a given set. If the set is an even amount of numbers, the middle two #'s are averaged.

To change mixed fractions into single fractions:

Multiply whole # by denominator & adding the numerator. (Ex: 2 1/4 => 9/4)

Scalene:

No equal angles.

Multiplication key words:

Of, times, area, & product.

Obtuse Triangle:

One angle measures greater than 90 degrees.

Order of Operations PEMDAS acronym:

Parentheses/Exponents/Multiply/Divide/Add/Subtract

Roots:

Root of a Product: n*Sqrt(a*b) = n*Sqrt(a) * n*Sqrt(b) Root of a Quotient: n*Sqrt(a/b) = n*Sqrt(a)/n*Sqrt(b) Fractional Exponent: n*Sqrt(a^m) = a^(m/n)

Polygon Angle Principle:

S=sum of interior angles of a polygon with n-sides. S=(n-3)180

Adjacent Angles:

Share a side and a vertex.

Tangent:

Something that touches a circle at only on point without crossing through it.