# Complex Numbers 3.3 Perform arithmetic operations on complex numbers

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Complex Numbers 3.3 Perform arithmetic operations on complex numbers
Solve quadratic equations having complex solutions Copyright © 2010 Pearson Education, Inc.

Properties of the Imaginary Unit i
Defining the number i allows us to say that the solutions to the equation x2 + 1 = 0 are i and –i. Copyright © 2010 Pearson Education, Inc.

Complex Numbers A complex number can be written in standard form as a + bi where a and b are real numbers. The real part is a and the imaginary part is b. Every real number a is also a complex number because it can be written as a + 0i. Copyright © 2010 Pearson Education, Inc.

Imaginary Numbers A complex number a + bi with b ≠ 0 is an imaginary number. A complex number a + bi with a = 0 and b ≠ 0 is sometimes called a pure imaginary number. Examples of pure imaginary numbers include 3i and –i. Copyright © 2010 Pearson Education, Inc.

The Expression If a > 0, then

Example 2 Simplify each expression. Solution

Example 3 Write each expression in standard form. Support your results using a calculator. a) (3 + 4i) + (5  i) b) (7i)  (6  5i) c) (3 + 2i)2 d) Solution a) (3 + 4i) + (5  i) =  i  i = 2 + 3i b) (7i)  (6  5i) = 6  7i + 5i = 6  2i Copyright © 2010 Pearson Education, Inc.

Solution continued c) (3 + 2i)2 = (3 + 2i)(3 + 2i)

Quadratic Equations with Complex Solutions
We can use the quadratic formula to solve quadratic equations if the discriminant is negative. There are no real solutions, and the graph does not intersect the x-axis. The solutions can be expressed as imaginary numbers. Copyright © 2010 Pearson Education, Inc.

Example 4a Solve the quadratic equation
Support your answer graphically. Solution Rewrite the equation: a = 1/2, b = –5, c = 17 Copyright © 2010 Pearson Education, Inc.

Example 4a Solution continued
The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary. Copyright © 2010 Pearson Education, Inc.

Example 4b Solve the quadratic equation x2 + 3x + 5 = 0. Support your answer graphically. Solution a = 1, b = 3, c = 5 Copyright © 2010 Pearson Education, Inc.

Example 4b Solution continued
The graph does not intersect the x-axis, so no real solutions, but two complex solutions that are imaginary. Copyright © 2010 Pearson Education, Inc.

Example 4c Solve the quadratic equation –2x2 = 3.
Support your answer graphically. Solution Apply the square root property. Copyright © 2010 Pearson Education, Inc.

Example 4c Solution continued
The graphs do not intersect, so no real solutions, but two complex solutions that are imaginary. Copyright © 2010 Pearson Education, Inc.

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