Download 7.8 Applications of Quadratic Equations 8.1 Rational

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=300 + 500x -100x2. 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=1,300 -100p (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
• a2+b2=c2
c
a
b
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
P
where P and Q are both POLYNOMIALS
Q
Examples:
2x  3
,
x5
x  5x  6
,
2
x 1
2
a b
ba
• 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
x  81
x 9
2
• 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
2
x 4
2
x  2x  8
Reduce
r  5r  6
2
r 1
2
2 w  16
2
4 w  12 w  8
3
x  27
2
3 x  3 x  36
3
xa  3 x  4a  12
xa  2a  3 x  6
• 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.
3 a
a 3
2x  8
2 x
2
Definition of Rational Functions
• A rational function is any function that can be
written in the form
P( x)
f ( x) 
Q( x)
• Where P(x) and Q(x) are both polynomials and
Q(x) cannot equal 0.
Evaluate a Rational Function
x5
If f ( x) 
find f (5).
x 3
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
2x  4
f ( x) 
,
x  12
4 x  15
g ( x) 
,
x2
2
4
h( x )  2
x  16
Difference Quotients
f ( x)  f (a)
Slope of line through PQ 
xa
• 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.
f ( x)  f (a)
If f ( x)  3x  5, find
xa
f ( x)  f (a)
If f ( x)  4 x  2, find
xa
f ( x)  f (a)
If f ( x)  2 x  5, find
xa
2
• 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.
f ( x  h)  f ( x )
Slope of line through PQ  m 
h
f ( x  h)  f ( x )
If f ( x)  4 x  2, find
h
f ( x  h)  f ( x )
If f ( x)  x  4, find
h
2