Download Lesson7.5

Lesson 7.5

We have studied several ways to solve
quadratic equations.
◦ We can find the x-intercepts on a graph,
◦ We can solve by completing the square,
◦ We can use the quadratic formula.
y  x 2  6x  8
y

y   x  3  1
( 6)  ( 6)2  4(1)(8)
x 
2(1)
2

x

y  x 2  6x  9   8  9 
b  b 2  4ac
x 
2a









y  x 2  6x  8
x  2, 4
y  1x 2  6x  8
1   x  3
2
x  3  1
x  31
x  4,2

6  36  32
2
6 4
x 
2
x  4,2
x 

What happens if you try to use the quadratic
formula on an equation whose graph has no
x-intercepts?
y  1x 2  4x  5
y
b  b 2  4ac
x 
2a



x










4  42  4(1)(5)
x 
2(1)
4  16  20
2
4  4
x 
2
4  4 4  4
x 
,
2
2
x 

The graph of y=x2+4x+5 at right
shows that this function has no
x-intercepts. Using the quadratic
formula to try to find xintercepts, you get
y  1x 2  4x  5
x 


4  4 4  4
,
2
2
How do you take the square root
of a negative number?
They are nonreal, but they are
still numbers.
4  4
4  4
and
2
2
are nonreal numbers

You are familiar with square roots of positive
numbers
16,
25,
16  4,

25  5,
15
15  3.873
But we can also have square roots of negative
numbers.
16, 25, 15

Numbers that involve the square roots of
negative numbers are called complex
numbers.
 To express the square root of a negative
number, we use an imaginary unit called i.
 We start by defining
i 2  1 or i 
 We can rewrite
4 1  2i
 Therefore,
4 
4  4
4  2i

2
2
and
4  4
4  2i

2
2
1
 These two solutions are a conjugate pair.
 one is a +bi and
2  i
 the other is a- bi. 2  i
 The two numbers in a conjugate pair are
called complex conjugates.
 Why will complex solutions to the quadratic
formula always give answers that are a
conjugate pair?
b  b 2  4ac
x 
2a
Remember that a complex number will be a
number in the form a + bi, where a and b
are real numbers and
i  1
3  4  3  2i
9  5  9  i 5
2  9  2  3i
7  2  7  i 2












Solve x2+3=0.

 


y
x



Checking our Solutions
The two imaginary numbers are solutions to the original
equation, but because they are not real numbers, the graph
of y =x2+3 shows no x-intercepts.


When computing with complex numbers,
there are conventional rules similar to those
you use when working with real numbers. In
this investigation you will discover these
rules. You may use your calculator to check
your work or to explore other examples.



Addition and subtraction
of complex numbers is
similar to combining like
terms (such as 2-4x and
3+5x).
You can use your
calculator to add
complex numbers such
as 2-4i and 3+5i
Change the setting to
Rectangular for Complex
numbers.

Make a conjecture about how to add complex
numbers without a calculator.

You remember how to multiply 3+3x and
3-2x using the rectangle method.
2
2
3

3
x
3

2
x

9

9
x

6
x

6
x

9

3
x

6
x



3
3
9
- 2x
-6x
+ 3x
9x
2
-6x

Multiply these complex numbers. Express
your products in the form a +bi. Recall that
i2 =1.
2
+3i
3
-2i

Recall that every complex number a+bi has a
complex conjugate, a - bi. Complex conjugates
have some special properties and uses. Each
expression below shows either the sum or product
of a complex number and its conjugate. Simplify
these expressions into the form a+bi, and
generalize what happens.

Recall that you can create equivalent fractions
by multiplying the numerator and
denominator of a fraction by the same
quantity. For example,
3  3   k  3k
    
4  4   k  4k
 3  2  3 2

 
 
2
2  2  2 
3

You will use a similar technique to change the
complex number in each denominator into a
real number. Use your work from Part 3 to
find a method for changing each denominator
into a real number. (Your method should
produce an equivalent fraction.) Once you
have a real number in the denominator,
divide to get an answer in the form a+bi.

You cannot graph a complex number, such as
3+ 4i, on a real number line, but you can graph it
on a complex plane, where the horizontal axis is
the real axis and the vertical axis is the imaginary
axis. In the graph, 3+4i is located at the point with
coordinates (3, 4). Any complex number a+bi has
(a, b) as its coordinates on a complex plane.