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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.

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**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.

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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.

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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.

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**The Expression If a > 0, then**

Copyright © 2010 Pearson Education, Inc.

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**Example 2 Simplify each expression. Solution**

Copyright © 2010 Pearson Education, Inc.

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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.

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

Copyright © 2010 Pearson Education, Inc.

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**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.

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**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.

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**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.

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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.

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**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.

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**Example 4c Solve the quadratic equation –2x2 = 3.**

Support your answer graphically. Solution Apply the square root property. Copyright © 2010 Pearson Education, Inc.

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**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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Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.

Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.

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