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A numerical operation can be described as an action or process used to solved a numerical problem. Adding and Subtracting Signed Numbers If you have two numbers x and y, the following rules apply when these numbers are being added or subtracted. Rule of Signs for Addition: (x – y)…
To ensure that arithmetic calculations are performed consistently, we must follow the order of operations. If an arithmetic expression contains brackets, exponents, multiplication, division, addition, and subtraction, we use the following procedure: Perform all operations inside a bracket first (the operations inside the bracket must be performed in proper order).…
In mathematics, you will have trouble understanding the material unless you clearly understand the language of the words that are being used. Here are some commonly used words and phrases that you need to know, and that will be repeated throughout the duration of this course. Integers The values: ……
Mean The mean is a number that shows the center of the data. Another word for mean is average and it is mathematically symbolized as x̄ (x-bar). However, when calculating the mean of the whole population, you use the Greek letter μ (mu) instead. To calculate the average of a sample, you sum all the observations,…
The standard deviation (SD) is a measure of the spread of the data (how far from the normal it is). A number such as the mean or the standard deviation may be found either for an entire population (symbolized as σ or σx) or for a sample (symbolized as s) drawn from that…
In statistics, an entire group of people or things is called a population or universe. A population can be infinite or finite. Examples of finite populations include the number of airplanes owned by an airline, or the potential consumers in a target market. Examples of infinite populations include the number of…
A literal equation is one in which some or all of the constants are represented by letters. Arguably any mathematical formula expressing an actual relationship between its variables is a literal equation. Take the Pythagorean theorem formula as an example. null It consists of three variables, a and b are the…
Depending on the number of terms in the equation, these questions can go from being simple to extremely difficult. Easiest of these types is when you have a single term on the left side and a single term on the right. This of course was discussed earlier in this unit…
A key component to multiplying and dividing algebraic fractions is knowing how to do it to ordinary fractions. That being said, you’re first expected to review how it’s done before continuing. If after a few examples you feel confident enough, you may skip it. Multiplying Algebraic Fractions Just as you…
The skills you’ll learn in this lesson will come in handy unexpectedly one day when you’re stuck trying to simplify what appears to be an impossible expression to reduce. Take a look at the three expressions below: null null null At first glance, you might be questioning how do I…
Although this concepts was briefly discussed in one of the earlier lessons, you learned that sometimes you may need to combine several of the techniques to factor a single expression. As a result, to factor the expressions found in this section, you’ll have to device a plan before starting because…
Fractions always have a tendency to scare math student no matter what level of study they’re in. You’re likely to have been first introduced to fractions in elementary school, so before we start mixing in variables into our questions – as you’d expect with algebraic expressions – a good place to…
Generally, the higher the degree of a polynomial, the harder it becomes to factor. The highest degree you’re expected to factor in this course are cubic equation (those raised to the power of three). Specifically, we’ll look at examples similar in structure to quadratics that are a difference of square,…
In this unit’s final lesson, we’ll learn how to quickly factor general form quadratics that are considered “perfect square trinomials” (PST). In a PST, the first and last term of these trinomials are always perfect squares. If you don’t recognize the pattern of a PST, you could still factor the quadratic…
When a general form quadratic has an a coefficient greater than 1, the trial-and-error method no longer works. Take, for example, the equation: y = 3x² + 5x + 6 You can’t choose 3 and 2 as factors that multiply to 6 and add to 5 – it doesn’t work that way.…
For the first time in this course, you’ll learn how to convert a quadratic that’s in its general form to a quadratic in factored form. General form: y = ax² + bx + c → Factored form: y = a(x – r)(x – s) where r and s represent the…
So far we’ve learned three factoring techniques. The first one, common factoring, is a technique that can be used for any polynomial. The other two were specific for quadratics, namely trial-and-error and decomposition. The whole purpose behind factoring any quadratic – if you haven’t discovered already – is convert it in…
Another major part of algebra and converting quadratics into different forms is the ability to common factor. Think of this as the opposite of “expanding”, which is what we did in in the previous unit. When you factor an expression, you’re making it more condensed. Let’s start with what factor means. Let’s…
Graphing a quadratic function without a table of values isn’t as simple as linear functions. This process involves knowing the parabola’s direction of opening, and finding its solutions (also called roots or x-intercepts), vertex point, and y-intercept. These features are shown below: Sketching Quadratics with Roots Recall that to graph a quadratic…
The grouping of algebraic terms using parentheses plays a major role in how an expression gets evaluated. Symbols of grouping used in mathematical expressions include parentheses ( ), brackets [ ], and braces { }, and they all serve the same common purpose, that is, the terms they enclose are to…