1. Adding Radicals
1.1. Can only add like terms
1.2. Negatives can be removed by multiplying the radicle by "i "
1.3. Turn the radicles into like terms by dividing the term under the radicle by a perfect square and multiply the quotient by the square root of the perfect square that acted as the divisor
2. Imaginary Operations
2.1. Addition and Subtraction of Complex Numbers
2.1.1. Distribute the negative/positive to the terms
2.1.1.1. Remove the parentheses and add the terms together
2.2. Multiplication of Complex Numbers
2.2.1. Use the "double distribution method"
2.2.1.1. Simplify using the following rules
2.2.1.1.1. i^0=1
2.2.1.1.2. i^1=i
2.2.1.1.3. i^2=-1
2.2.1.1.4. i^3=-i
2.2.1.1.5. To find the exponent of "i " divide the original exponent by 4 and place the remainder as the new exponent
2.3. Simplifying a fraction with a denominator that is a complex number
2.3.1. Multiply the numerator and denominator by the conjugate of the dominator (flip the polarity of the second term's sign)
2.3.1.1. Simplify
3. Finding the Equation from given roots
3.1. Plug in the correct values into this equation: 0=x^2-(sum of roots * X)+ products of roots
4. Converting Quadratic Form to Vertex Form
4.1. Quadratic Form: ax^2 + bx + c = y
4.2. Vertex Form y=a(x-k)+h
4.3. Step 1: Sub in the a in the quadratic form for the a in vertex form
4.3.1. Step 2: Using the expression "-b/2a" use plug in the values of the quadratic to determine the x-coordinate of the vertex
4.3.1.1. Step 3: Set the answer from -b/2a equal to k
4.3.1.1.1. Step 4: Plug in the vertex's x coordinate into the quadratic and solve for the y coordinate of the vertex (set this equal to h)
5. Types of Numbers
5.1. Real: A combination of rational and irrational numbers, all non-imaginary numbers are real
5.1.1. Rational: Numbers that can be written in the form "a/b" this can include repeating decimals
5.1.1.1. Whole: The Natural numbers and 0
5.1.1.2. Natural: Integers that are greater than 1
5.1.2. Irrational: Numbers that can't be written in the format of "a/b" (ex: π, sqrt(2)
5.2. Imaginary: Numbers that don't fall in the category of Real
5.2.1. Pure Imaginary: Numbers that are in the form of bi where "i" is equal to sqrt(-1)
5.3. Complex Numbers: Numbers that include both real and imaginary components (ex: 4-3i)
6. Solving for X
6.1. Quadratic Formula
6.1.1. Requirements: Equation must be in the form of "ax^2 + bx + c = 0"
6.1.2. Formula: (-b ±sqrt(b^2-4ac))/(2a)
6.2. Factoring
6.2.1. Quadratic
6.2.1.1. Take out a GCF if possible and does not inhibit the factorability of the quadratic
6.2.1.1.1. Find the factors of "C" that when are plugged into (x±_)(x±_) yield the same equation as before
6.2.2. Cubics
6.2.2.1. Step 1: Factor out a GCF until the numbers left are perfect cubes
6.2.2.1.1. Step 2: Find the cube roots of the numbers left (Use A and B for cube roots in this example)
6.2.2.2. IS NOT USED FOR SOLVING X BUT RATHER STRICTLY FACTORING
6.2.3. Factor by Grouping
6.2.3.1. Make sure the equation have a total of 4 terms
6.2.3.2. Find a common factor between the first 2 terms
6.2.3.2.1. Find a common factor between the second two terms
7. Finding the number and nature of the roots
7.1. Plug the values for "a" "b" and "c" into b^2-4ac (The Discriminant)
7.1.1. If the discriminant is greater than 0 then there are two roots that are real and unequal
7.1.1.1. If the discriminant is a perfect square, then the roots are rational
7.1.1.2. If the discriminant is not a perfect square, then the roots are irrational
7.1.2. If the discriminant is 0 then there is 1 real and rational root
7.1.3. If the discriminant is less than 0, then there are 2 unreal solutions that are unequal
8. Finding a Parabola's Equation when given the points
8.1. Plug all of the points into the equation of y=ax^2+bx+c
8.1.1. This creates an amount of equations equal to the amount of points
8.1.2. Mathematical Method
8.1.2.1. Use the "Elimination Method" in order to solve the system of equations
8.1.2.1.1. Once one value is found, it can be plugged in to find the other values
8.1.3. Calculator Method
8.1.3.1. Plug the equations into a matrix in the form of ax^2+bx+c=0
8.1.3.1.1. Use the "rref" function on the matrix created to find the a, b, and c values of the parabola's equation