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SECTION 2.7 COMPLEX ZEROS OF A QUADRATIC FUNCTION COMPLEX ZEROS OF A QUADRATIC FUNCTION

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SQUARE ROOTS OF NEGATIVE NUMBERS Is a value we have dealt with up to now by simply saying that it is not a real number. And, up to now, we have dealt with the following equation by simply saying it has no solution: x 2 + 4 = 0

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DEFINITION OF i i 2 = - 1 The number i is called an imaginary number. Imaginary numbers, along with the real numbers, make up a set of numbers known as the complex numbers.

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COMPLEX NUMBERS Imaginary Real 5-1 1/2.7 i2i - 3i2/3i - 3i2/3i 4 + 5i -7 + i 1/2 + 3/4i

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COMPLEX NUMBERS All numbers are complex and should be thought of in the form: a + bi Real Part Imaginary Part

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COMPLEX NUMBERS a + bi Real Part Imaginary Part When b = 0, the number is a real number. Otherwise, the number is imaginary.

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OPERATING ON COMPLEX NUMBERS Addition: Example: (3 + 5i) + ( - 2 + 3i) Subtraction: Example: (6 + 4i) - ( 3 + 6i)

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OPERATING ON COMPLEX NUMBERS Multiplication: Example: (5 + 3i) (2 + 7i) Example: (3 + 4i) ( 3 - 4i)

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CONJUGATES 2 + 3i = 2 - 3i Multiplying a complex number by its conjugate always yields a nonnegative real number.

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THEOREM: If z = a + bi z z = a 2 + b 2

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Writing the reciprocal of a complex number in standard form. Example:

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Writing the quotient of complex numbers in standard form. Example:

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Example:

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POWERS OF i i 1 = i i 2 = - 1 i 3 = - i i 4 = 1 i 5 = i and so on

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QUADRATIC EQUATIONS WITH A NEGATIVE DISCRIMINANT Quadratic equations with a negative discriminant have no real solution. But, if we extend our number system to the complex numbers, quadratic equations will always have solutions because we will then be including imaginary numbers.

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EXAMPLE

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Solve the following equations in the complex number system: x 2 = 4x 2 = - 9

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WARNING!

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EXAMPLE Solve the following equation in the complex number system: Solve the following equation in the complex number system: x 2 - 4x + 8 = 0

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DISCRIMINANT If b 2 - 4ac > 0 Two unequal real solns If b 2 - 4ac = 0 One double real root If b 2 - 4ac < 0 Two imaginary solutions

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EXAMPLE: Without solving, determine the character of the solution of each equation in the complex number system: 3x 2 + 4x + 5 = 0 2x 2 + 4x + 1 = 0 9x 2 - 6x + 1 = 0

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CONCLUSION OF SECTION 2.7 CONCLUSION OF SECTION 2.7

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