**Operations**

**One Variable**

– Like terms: Terms whose variables (such as x or y) with any exponents (such as the 2 in x2) are the same. Examples: 7x and 2x are like terms because they are both “x”.

– REMINDER: Always check what they are asking for before selecting your answer. For example, the questions may ask “what is x + 4” as opposed to just “what is x”.

**Two or More Variables**

– 3 Quadratic Identities (unfactored to factored form)

(x2-y2)=(x+y)(x-y)

x2+2xy+y2=(x+y)2

x2-2xy+y2=(x-y)2

**Variables in Choice**

**System of Equations**

METHOD 1: Substitution

2x + 4y = 6 AND y = 5 —–> 2x + 4(5) = 6

METHOD 2: Elimination

2x + 4y = 6

Subtract these two equations to eliminate the “x”:

2x – y = – 9

5 y = 15

y = 3

METHOD 3: Set Equal

y = 2x + 3 AND y = 4x + 1 ——> 2x + 3 = 4x + 1

**Absolute Value**

Break the equation into two equations (positive and negative)

Absolute Value Equations will always have two answers!

| 2x + 4 | = 6

2x + 4 = + 6 2x + 4 = – 6

2x = 2 2x = – 10

x = 1 x = -5

**Inequalities**

– Pretend the inequality sign is an equal sign and solve for the variable!

– When you divide or multiply by a negative, FLIP THE SIGN!

-6x – 9 > 3

Add 9

-6x > 12

Divide by -6 AND flip the sign

x < -2

**Exponents**

**Functions**

– The “x” is Just a Place-Holder! It is just there to show us where the input goes and what happens to it. It could be anything!

**Sequences**

– Sequences are all about finding the pattern!

**Quadratics**

ax^2 + bx + c = 0

METHOD 1: Factoring

– Find two numbers that multiply to the last term and add to the middle term

METHOD 2: Quadratic Formula

The Discriminant

– The discriminant is everything under the radical in the quadratic equation = b^2 – 4ac

– If the discriminant is POSITIVE, then there are 2 real roots (“roots” is another word for “solutions” when equations are written in ax^2+by+c = 0 form).

– If the discriminant is ZERO, then there is 1 real root.

– If the discriminant is NEGATIVE, then there are no real roots. (#13 C Test 6)

– Know how to complete the square

**Defined Operations**

**Imaginary Numbers**

– Binomial addition involving constants and i by combining like terms (adding and subtracting complex numbers)

– Multiplying by the conjugate of the denominator with complex numbers (#11 Test 2)

**Number Theory**

– Integers: whole numbers, including zero and negative whole numbers

– Prime Numbers: positive integers that are only divisible by themselves and the number 1 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, etc…

**Divisibility & Remainders**

– Polynomial Remainder Theorem (#29 C Test 1) (#7 NC Test 3)

– Divisible by 2 = last digit 0, 2, 4, 6, 8?

– Divisible by 3 = sum of digits divisible by 3?

– Divisible by 4 = last 2 digits divisible by 4?

– Divisible by 5 = last digit 0 or 5?

– Divisible by 6 = follow rule 2 and 3?

– Divisible by 8 = last three digits divisible by 8?

– Divisible by 9 = sum of the digits divisible by 9?

– Divisible by 10 = last digit 0?

**Ratio & Proportions**

– Direct and Indirect Proportion (a1/b1)=(a2/b2) and (a1a2 = b1b2), respectively

**Percents**

– Percentage = (Part/Whole)

– Percent Change = (Difference/Original) x 100)

**Triangles**

– Special Triangles (3/4/5, 5/12/13, 30/60/90, 45/45/90)

– Pythagorean theorem is used to find the third side of a right triangle. It is ONLY for right triangles. NOTE: Hypotenuse is always the longest side.

– The largest side is opposite the largest angle. The smallest side is opposite the smallest angle.

– All three angles add up to 180 degrees!

– The sides of similar triangles all have the same respective proportions. (#17 NC Test 1, #18 NC Test 2)

– The Third Side Rule for Triangles (a-b) < c < (a+b) if c represents the “third side” and b and a represent the lengths of the other two sides.

– The proportion of distance that you travel along the hypotenuse of a right triangle is equal to the proportion of distance that you travel along both legs. (#16 NC Test 4)

**Circles & Sectors**

– The Circle Proportionality Formula (Slice/Area = Arc/Circumference = Measure of Inner Angle/360)

– The arc measure formed by an angle with its vertex on a circle is double the measure of the angle. (#36 C Test 5)

– The equation of a circle with center (h,k) and radius r: (x-h)2 + (y-k)2= r2 (#24 C Test 1)

**2D & 3D Objects**

Surface Area of a Cube = 6s2

Area of a triangle = 1/2 ab sin C

– Unfortunately all the formulas below are not included on the ACT, unlike on the SAT.

– The vertex of a parabola is located at the midpoint of its x-intercepts (#12 NC Test 3)

– The vertex (h,k) form of a parabola: a(x-h)^2 + k

– When an upward projectile reaches its highest point, its velocity is zero.

– When an upward projectile lands, its height is zero.

– To find the intersections of two lines, set them equal to one another (#13 test 4)

– In a system of linear equations, there is no solution if the slopes of the two lines are the same (parallel) and the y-intercept is different. (see #9 Test 3) Conversely, there are infinitely many solutions is the slopes of the two lines are the same and the y-intercept is also the same (#20 NC Test 2)

– The “zeroes” or “roots” of a function are the x-coordinates where it crosses the x-axis (and where the y value outputs zero).

– A polynomial of Nth degree has at most N-1 changes in direction.

– Domain and Range

– Parallel Lines and Transversals (#36 Test 1)

– The Formula for a Line (standard y=mx+b format as well as point-slope format: y-y1 = m(x-x1), and the slope equation (y2-y1) / (x2-x1) ).

– Positive and Negative Associations in Graphs (#5 C Test #1)

**Number Lines & Line Segments**

A **line** extends forever in both directions.

A **line** **segment** has two endpoints.

A **ray** is a part of a line that has one endpoint and goes on infinitely in only one direction.

A midpoint is the middle of the line. To solve for the midpoint, take the average of the x value and the average of the y value.

**Angles**

A line measures 180degrees.

Types of angles to know:

– opposite angles

– supplementary angles

– opposite interior angles

– complimentary angles

**Degrees & Radians**

– π radians = 180 degrees (#19 NC Test 2)

**Probability**

Probability = (Desired Possibilities / Total Possibilities)

**Permutations & Combinations**

Permutation = arranging the objects or numbers in order

Combinations = selecting objects or numbers from a group, in which order doesn’t matter.

**Simple & Compound Interest**

**Simple interest** is based on the principal amount of a loan or deposit. In contrast, **compound interest** is based on the principal amount and the interest that accumulates on it in every period.

**Rates & Work**

Distance = Rate x Time (#38 C Test 5, #9 C Test 3)

**Mean, Median, & Mode**

Mean: average = (Total / Number of things)

Median: middle number

Mode: most frequent number

Concept: Weighted Averages (#19 NC Test 5)

**Standard Deviation & Range**

**Tables & Charts**

– Box and whisker plots (showed up on March 2018 SAT)

**Matrices**

**Scientific Notation**

– Negative exponents move the decimal to the left.

– Positive exponents move the decimal to the right.

**Word Problems**

– Word problems can come in the form of any topic above. The best trick is to convert the words into a mathematical equation first!

**Other Helpful Study Materials**

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