7.8 Applications of Quadratic Equations 8.1 Basic Properties and Reducing to Lowest Terms

4 types of problems that we will address with quadratic equations This only skims the top of the possibilities that quadratic equations provide us. 1. Number problems 2. Geometry problems 3. Business problems 4. Pythagorean Theorem (right triangles)

Number Problems The product of two consecutive odd integers is 63. Find the integers. Integers are Z={…-3,-2,-1,0,1,2,3…} If we know one of the numbers what would the other be? How do you get that? So now lets assume we don’t know the first one, what should we let it be if we don’t know it? The second one should be found the same way as when we did know the first one. If they produce a product of 63 what should we do with the two numbers?

The product of two consecutive even integers is 48. Find the integers.

Sometimes these number problems will provide us other information such as a sum and a product of two numbers. We will identify the two numbers in regards to their sum, then use their product to set up an equation. Say we have two numbers whose sum is 30. What should we call one of those numbers? What about the other?

Example The sum of two numbers is 14. Their product is 48. Find the two numbers.

Example The sum of two numbers is 17 and their product is 60. Find the two numbers.

Geometry Problems Usually geometry problems deal with area. May have to recall a few area formulas. Suppose the length of a rectangle is 3 more than twice the width. The area of the rectangle is 44. Find the dimensions. First determine which dimension (length or width) that you know the least amount of information about, let that then be your simplest variable. Write the other dimension in reference to your first.

Example with triangle The height of a triangle is 3 inches more than the base. The area is 20 square inches. Find the base and height of the triangle.

Example The numerical value of the area of a square is twice its perimeter. What is the length of its side? This type of problems requires knowledge about the area of a square and the perimeter of a square. Let x= the length of a side.

Business Problems A company can manufacture x hundred items for a total cost of C= x -100x 2. How many items were manufactured if the total cost is $900? We have a cost and a cost function, set them equal and solve for x. Remember our answer will be in hundreds of items.

A manufacturer of small portable radios knows that the number of radios he can sell each week is related to the price of the radios by the equation x=1, p (x is the number of radios, p is the price per radio). What price should she charge for the radios to have a weekly revenue of $4,200? R=price*quantity. = p*x

The Pythagorean Theorem a 2 +b 2 =c 2 a b c The three sides of a right triangle are three consecutive integers. Find the lengths of the three sides. What can we call the first integer? How do you then find the next? And then the next?

The longer leg of a right triangle is 2 more than twice the shorter leg. The hypotenuse is 3 more than twice the shorter leg. Find the length of each side. Which side do we know the least amount of information about?

The hypotenuse of a right triangle is 10 inches, the lengths of the two legs are given by two consecutive even integers. Find the two sides, and then the area of the triangle.

8.1 Basic Properties and Reducing to Lowest Terms Rational numbers are any number that can be written as a/b, where b cannot equal zero. Basically a rational number is any number that can be expressed as a fraction. We will now deal with rational expressions.

By definition rational expressions are stated as anything written in the form where P and Q are both POLYNOMIALS Examples:

We may multiply the numerator and denominator by any value that we wish as long as we do it to both the top and bottom because it may change the form of the original expression but will produce an expression that is always equivalent to the original. You may do the same with division.

Reduce When dealing with the reducing process of rational functions, you must break down each polynomial into its factors so that multiplication is being shown between factors. The way it is written now as sums and differences does not allow for reduction/cancelling.

Reduce

On occasions we will come across factors that look like they may cancel, but they differ by signs. There is a trick to get them to cancel. The trick is to factor out a -1 out of either the top or bottom (not both) and this should then change the appearance so you can cancel.

Definition of Rational Functions A rational function is any function that can be written in the form Where P(x) and Q(x) are both polynomials and Q(x) cannot equal 0.

Evaluate a Rational Function We would be able to choose any value of x that we want to evaluate this function except one. If we were to substitute 3 what primary rule do we violate? So this being said x can be anything but the number 3. So the DOMAIN is all real numbers EXCEPT 3.

If the domain of a rational function is not specified, it is assumed to be all real numbers for with the function is defined. Which again is all real numbers that do not give you zero in the denominator. Find the domains for the following functions

Difference Quotients This is a calculus concept, but allows us to find the slopes of curves and not just lines, however we use lines to do it. The smaller the distance between x and a then we begin developing the exact slope of the curve at a point x.

This is an additional difference quotient, the only change is the way in which we identify or label the points. Here x is a variable and h is a constant.