# Quadratic Functions

**Example 2** Find the maximum area inclosed by 1000
yards of fence for six

rectangular corrals in the following configuration.

In this example there are four vertical fence lines and
three horizontal

fence lines. So we have the relation

4x + 3y = 1000

If we solve for y we have
Then the area would be given by

the function

Thus we see that the length of the vertical sides are 125
yards and the

maximum area would be square yards . The
length of the horizontal

sides would then be yards.

**§5.6 The Quadratic Formula **

If we consider the general quadratic function f(x) = ax^{2}
+ bx + c and

complete the square we compute the following

So the vertex is

Now if we set f(x) = 0 and solve for x we compute

The last equation is the **Quadratic Formula.**

**Example 1** Use the quadratic formula to find the
zeros of f(x) = 2x^{2}−3x+1.

Here we note that a = 2, b = −3 and c = 1, so we have

**Example 2** Use the quadratic formula to find the
zeros of f(x) = 2x^{2} + 1.

Here we note that a = 2, b = 0 and c = 1, so we have

Since there is no real number when squared will be
negative, we conclude

that this function has no real zeros. The graph of this function shows

clearly demonstrates that fact as well.

**§5.7 Complex Numbers **

Just because the function f(x) = 2x^{2} + 1. doesn’t have
any roots in

the Real numbers, does not mean it doesn’t have roots in some other set of

numbers.

Consider the following sequence of functions and their roots.

Lets begin by assuming that the only numbers that we know
are the

Natural or Counting numbers, {1, 2, 3, · · · }

Consider the function f(x) = x − 1, it’s zero is 1 which
is a counting

number.

Now consider the function f(x) = x + 1, it’s zero is −1,
which is not a

counting number. But if all we have are the counting numbers then we have

no root. So we enlarge the counting numbers to include the negative counting

numbers and zero, which we call the integers, {· · · ,−2,−1, 0, 1, 2, · · · }.
And

now we can have zeros to functions like f (x) = x + c where c is a counting

number.

Now consider the function f(x) = 2x − 1, it’s zero is 1/2
which is not an

integer! So again if all we have are the integers we do not have a root to this

function. So again we enlarge the integers to include all possible fractions ,

which we call the rational numbers , and p, q
are integers with no

common factors }. So again we can have zeros to functions like f(x) = ax+b

where a and b are integers.

OK, now consider the function f(x) = x^{2} − 2. It has two
zeros and

neither one is rational! So we must enlarge our set of numbers to include the

real numbers to solve equations such as this.

Let us now consider one more function f(x) = x^{2} + 1. If
it has zeros,

they would have to be And neither one of
these numbers could

possibly be real. So we must again enlarge our set of numbers to include

these possible roots.

**Definition** We say
We do this mostly for convenience, so we
don’t

have to write over and over again.

**Definition** We say that a complex number is a number
of the form a + bi

where a and b are real numbers. We call a the real part of the complex number

and b is the imaginary part of the real number. So a complex number is the

formal sum of a real number and the product of a real number times

If the real numbers are represented by the real number
line, then the

complex numbers (where b ≠ 0) cannot be on the real number line. So where

are they?

We note that i is not real, so we place it one unit above
0 on the x-axis,

and we draw a second copy of the real numbers perpendicular through the

origin and through the point labelled i, and we label the corresponding points

on the vertical axis with multiples of i .

Now every point in the plane formed by these two axis
represents a com-

plex number, where the real part is indicated by its horizontal distance from

the vertical axis and the imaginary part is indicated by its vertical distance

from the horizontal axis. For example the points 2 + 2i, −1 + i, and 2 − 3i

are indicated on the graph below.

**§5.8 Complex Arithmetic**

The general strategy of arithmetic with complex numbers is
to treat the

complex numbers as binomials, and then do binomial arithmetic remember-

ing that

Let a + bi and c + di be complex numbers, e.g. 2 + 3i, and 1 − 4i

(a + bi) + (c + di) = (a + c) + (b + d)i,

e.g.

(2 + 3i) + (1 − 4i) = 3 − i.

(a + bi) − (c + di) = (a − c) + (b − d)i,

e.g.

(2 + 3i) − (1 − 4i) = 1 + 7i.

MULTIPLICATION

(a + bi) · (c + di) = a · c + a · di + c · bi + b · d · i^{2}

ac + bd(−1) + (ad + bc)i = (ac − bd) + (ad + bc)i

e.g.

(2 + 3i) · (1 − 4i) = 2 · 1 − 3 · (−4) + (3 − 8)i = 14 − 5i.

**DIVISION**

Division is a little trickier, but still not too bad.

e.g.

Now we look at the quadratic formula
and recognize

that when b^{2} − 4ac is positive we will have two real roots, but if b^{2} − 4ac is

negative we will have two non-real roots of the form A+Bi and A−Bi, where

The remaining case is where b^{2} − 4ac = 0,

in this case there is one root, The expression b^{2} − 4ac is called the

discriminant since it discriminates between the natures of the roots of the

quadratic function.

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