Unit 3: Equations, Inequalities, and Variations

The absolute value of a number is defined as its distance from zero on a number line. Written as a value between two vertical lines, there are several basic properties of absolute values:

• |a| = |-a|
• |a - b| = |b - a|
• |a| * |b| = |ab|
• |a| / |b| = |a/b|

Solving equations with absolute values is very similar to solving any other equations, except that absolute values yield two possibilities. In order to remove the absolute value sign, you must set up two equations; one for the value inside the absolute value sign, and one for its negative. Hence, |a| = 5 will yield two answers: 5 and -5. However, always check for extraneous solutions by plugging both answers back into the original equation (especially the negative solution).

Absolute values in inequalities is also a relatively simple concept. Also take the value within the absolute value symbol and its negative, and create two equations for two solutions. If |b+2| < 6, then b+2 < 6, making b < 4, and -b - 2 < 6, making b > -8.

Quadratic inequalities are more complicated than normal inequalities. First off, the quadratic should be factored.
Next, the critical values (values for x at which the quadratic equals 0) should be found.
Several ranges, written in interval notation, should be written from negative infinity to the lowest critical value, then from that value to the next critical value, and so on until you reach the highest value to infinity. Whether or not the critical values should be included in the intervals depends on whether the quadratic is greater/less than and/or equal to.
Using a chart, you need to determine whether any given value in an interval will yield a negative or a positive result. This should be done for all intervals. The intervals that corrispond to your original inequality will represent the range(s) of your solutions.

Rational expressions can usually be simplified by cancelling out like terms. It is also necessary to determine the domain by finding values that make the expression invalid, and disregarding these values in your solution.

Rational equations involve two rational equations that are equivalent to each other. Simply cross multiply or find the Least Common Denominator (LCM) and then solve as you would a normal equation. However, watch out for invalid solutions by discluding any impossible values (i.e. negatives under a square root).

Rational inequalities are similar to quadratic inequalities in that they also involve critical values. In order to determine the critical values, both the numerator and denominator of hte rational expression must be factored, and each of these factors must be set to zero. From there, you can solve for your variable as you would in a quadratic inequality. Always remember to state your domain restrictions.

Direct variation refers to a proportion between an x value and a y value, such that x/y always equals a constant.

Inverse variaiton is similar to direct variation, except that xy always equals a constant, so x decreases as y increases.

Unit 2: Factoring Polynomials

There are several methods to factoring polynomials, and several types of polynomials. Most, however, come as quadratic equations in the form ax²+bx+c.

Factoring quadratic equations with a leading (a) coefficient of 1 is relatively simple. When factored, the polynomial will be expressed as a binomial.
The quadratic involves two operations; starting off, you need to observe the second one. If it is negative, then the factored quadratic will be written as (x + __)(x - __). If it is positive, then you must observe the first operation, which will determine the operation defined in both factors of the factored quadratic, either (x + __)(x + __) or (x - __)(x - __).
After determining how the quadratic will be expressed, you need to find two factors of the third coefficient (c) that add or subtract to the middle coefficient (b), depending on the quadratic.
When you determine these two factors, simply place them in their corresponding places, and you will have successfully factored the quadratic.

Factoring quadratics with a leading coefficient (a) that is not 1 is somewhat more complicated. First off, you need to multiply the leading coefficient with the last coefficient (c) in order to yeild the master product.
Find two factors of the master product that add to the middle coefficient (b), and place these two factors in their corresponding places.
The result will have an extra factor of the leading coefficient (a), so divide it out and simplify. 

If a binomial represents the difference of two squares, then a² – b² = (a + b)(a – b).

A quadratic with two equal factors can be factored in two ways:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

If a quadratic involves higher powers, then z-substitution may be used.
ex. if z = x², then x⁴ – 6x² + 8 = z² – 6z + 8

The sum or differene of 2 cubes can be expressed in two ways:

• a³ + b³ = (a + b)(a² – ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

Unit 1: Operations

Radicals
Radicals are a fun concept, but in order to understand them, you must know a few fundamental properties.
So, here's a few of properties you should memorize:

• ⁿ√(ab) = ⁿ√(a) x ⁿ√(b)
• ⁿ√(a/b) = ⁿ√(a) / ⁿ√(b)
• ⁿ√(a) = a^1/2
• ^m√(ⁿ√(a)) = ^mn√(a)

You can use these properties to simplify radicals, or solve radical equations.

As with all mathematical elements, radicals have parts. The radical itself can be of multiple powers (i.e. square roots, cube roots), so the root that you are trying to find is called the index. The value under the radical symbol is called the radicand.

In order to add or subtract radicals, their radicands must be the same. Add or subtract the coefficients, and keep the common radicand in order to find the sum or difference. Remember, you can never add or subtract radicals with different radicands.

In order to multiply or divide radicals, use the above properties. Multiplying the coefficients yeilds the resulting coefficient. As for the radical part, they remain seperate unless they share an index. You can always combine radicals with the same index, but you cannot place a square root and a cube root under the same radical.

Solving equations with radicals involves using any of the above properties to find a variable.  Remember, you must always follow these rules, or Mr.Sihler will throw you into math jail. When you find a result, always check for extraneous, or invalid, solutions. Always, always, always plug your answers back into the original equation to make sure that they are valid. Often, radicals create extra solutions that do not exist.

Operations with Complex Numbers and i
Complex numbers can often be confusing and difficult to understand, but the basic idea originates from the concept of the square roots of negative numbers. Complex numbers involve both real numbers, and imaginary ones. Imaginary numbers share a common factor, i, which represents the square root of -1. i and its multiples must always be treated as seperate from the real portion of the complex number, similarly to radicals.

Adding or subtracting complex numbers is a relatively simple concept. Real numbers can be added together, and like radicals, imaginary numbers can also be grouped together. Simply combine the coefficients, and leave i as it is. Remember that complex numbers are still numbers on their own, and must be treated as such.

Multiplying complex numbers is slightly more complicated. You can use the distributive property to combine real and imaginary numbers, but you must then simplify. Multiplying real and imaginary numbers is similar to multiplying with radicals; simply multiply with the imaginary number's coefficient. However, when multiplying imaginary numbers with imaginary numbers, you can eliminate the i, but the resulting number is always negative. Remember that i is the square root of -1, so multiplying i with itself will always yeild a negative number. The same applies to dividing complex numbers.

Powers of i repeat in a pattern:

i¹ = i

i² = 1

i³ = - i

i⁴ = -1

i⁵ = i

...

Negative and Rational Exponents
Exponents, like radicals, have several fundamental properties that are important to memorize: 

• (x^a)(x^b) = x^a+b
• (x^a)^b = x^ab
• x^-a = 1/x^a
• ^a√(x) = x^1/4

When solving an exponential equation, the rules above must be adhered to. However, you may find it necessary to convert exponents into another base (i.e.

4^{2 =} 2⁴). Simply rewrite the given base as an exponent of the desired base, and use the above properties to simplify.

Welcome

So, uh, I've thought about what I would do over the summer, and I decided to create this blog as a resource for anybody entering 9HT Math. Feel free to use this blog as a study guide, or just to browse before the next school year. If you have any questions, email me at freedomtower1@gmail.com. Enjoy :)