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

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Slide 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 x = 0 are i and –i.

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Slide 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.

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Slide 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.

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Slide Copyright © 2010 Pearson Education, Inc. The Expression If a > 0, then

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Slide Copyright © 2010 Pearson Education, Inc. Example 2 Simplify each expression. Solution

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Slide Copyright © 2010 Pearson Education, Inc. 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

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Slide Copyright © 2010 Pearson Education, Inc. Solution continued c) ( 3 + 2i) 2 = ( 3 + 2i)( 3 + 2i) = 9 – 6i – 6i + 4i 2 = 9 12i + 4( 1) = 5 12i d)

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Slide Copyright © 2010 Pearson Education, Inc. 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.

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Slide 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

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Slide 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.

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Slide Copyright © 2010 Pearson Education, Inc. Example 4b Solve the quadratic equation x 2 + 3x + 5 = 0. Support your answer graphically. Solution a = 1, b = 3, c = 5

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Slide 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.

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

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Slide 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.

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