he solutions to a quadratic equation can be found directly from the quadratic formula. The equation ax2 + bx + c = 0 has solutions The advantage of using the formula is that it always works. The disadvantage is that it can be more time-consuming than some of the methods previously discussed. As a general rule you should look…
A system of equations refers to a number of equations with an equal number of variables. We will only look at the case of two linear equations in two unknowns. The situation gets much more complex as the number of unknowns increases, and larger systems are commonly attacked with the aid of a computer. A…
The whole problem with solving a system of equations is that you cannot solve an equation that has two unknowns in it. You need an equation with only one variable so that you can isolate the variable on one side of the equation. Both methods that we will look at are techniques for eliminating one…
Another method for factoring these kinds of quadratic trinomials is called factoring by grouping. Factoring by grouping can be a bit more tedious and is often not worth the trouble if you can find the correct factors by some quick trial and error. However, it works quite well when the factors are not immediately obvious,…
The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers. Natural Numbers or “Counting Numbers” 1, 2, 3, 4, 5, . ….
Solving a quadratic (or any kind of equation) by factoring it makes use of a principle known as the zero-product rule. Zero Product Rule If ab = 0 then either a = 0 or b = 0 (or both). In other words, if the product of two things is zero then one of those two things must be zero, because the only way…
General Strategy Recall the general strategy for setting up word problems. Refer to the Problem Solving Strategies page for more detail. Read the problem carefully: Determine what is known, what is unknown, and what question is being asked. Represent unknown quantities in terms of a variable. Use diagrams where appropriate. Find formulas or mathematical relationships…
1. Remove Common Factors if possible 2. If the coefficient of the x2 term is 1, then x2 + bx + c = (x + n)(x + m), where n and m Multiply to give c Add to give b 3. If the coefficient of the x2 term is not 1, then use either 1. Guess-and Check List the factors of the coefficient of thex2 term List the factors of the constant term Test all the possible…
Canceling Like Factors When we reduce a common fraction such as we do so by noticing that there is a factor common to both the numerator and the denominator (a factor of 2 in this example), which we can divide out of both the numerator and the denominator. We use exactly the same procedure to…