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,

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.

2

2

2

2

6 8

6 9 8 9

3 1

1 3

3 1

3 1

4,2

y x x

y x x

y x

x

x

x

x

2 6 8

2,4

y x x

x

x

y

2

2

2

42

( ) ( )

1 6 8

6 6 1 84( )( )2(1)

6 36 322

6 42

4,2

y x x

b b acx

a

x

x

x

x

What happens if you try to use the quadratic formula on an equation whose graph has no x-intercepts?

x

y

2

2

2

42

4( )( )2( )

4 16 202

4

1

42

4 4 4 4,

4 5

4 4

2

1

2

51

y x x

b b acx

a

x

x

x

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 x-intercepts, you get

How do you take the square root of a negative number?

They are nonreal, but they are still numbers.

4 4 4 4and

2 2are nno umnreal bers

21 4 5

4 4 4 4,

2 2

y x x

x

You are familiar with square roots of positive numbers

But we can also have square roots of negative numbers.

Numbers that involve the square roots of negative numbers are called complex numbers.

16, 25, 15

16 4, 25 5, 15 3.873

16, 25, 15

To express the square root of a negative number, we use an imaginary unit called i.

We can rewrite 4 4 1 2i

4 4 4 22 2

and

4 4 4 22 2

i

i

Therefore,

2 1 o 1r ii We start by defining

These two solutions are a conjugate pair. one is a +bi and the other is a- bi.

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?

2 i

2 i

2 42

b b acx

a

Remember that a complex number will be a number in the form a + bi, where a and b

are real numbers and

1i

3 4 3 2i 2 9 2 3i

9 5 9 5i 7 2 7 2i

Numbe

rs y

ou

have

stu

died

Solve x2+3=0.

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.

Checking our Solutions

x

y

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.

3 + 3x

3 - 2x

9 9x

-6x -6x2

2 23 3 3 2 9 9 6 6 9 3 6x x x x x x x

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 34 4 4

k kk k

3 3 2 3 222 2 2

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.