Understanding Complex Zeros of Quadratic Functions and the Role of Imaginary Numbers
This section explores complex zeros of quadratic functions, particularly focusing on the implications of negative discriminants. A quadratic equation can possess solutions in the complex number system even when no real solutions exist. We define imaginary numbers and their role in forming complex numbers, which are expressed in the form a + bi. The section also outlines operations on complex numbers, including addition, subtraction, multiplication, and division, and discusses the significance of conjugates. Examples illustrate solving quadratic equations in the complex system.
Understanding Complex Zeros of Quadratic Functions and the Role of Imaginary Numbers
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Presentation Transcript
SECTION 2.7 • COMPLEX ZEROS OF A QUADRATIC FUNCTION
Is a value we have dealt with up to now by simply saying that it is not a real number. SQUARE ROOTS OF NEGATIVE NUMBERS And, up to now, we have dealt with the following equation by simply saying it has no solution: x2 + 4 = 0
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 complexnumbers.
COMPLEX NUMBERS Imaginary Real i 2i - 3i 2/3i 4 + 5i -7 + i 1/2 + 3/4i 5 -1 1/2 .7
COMPLEX NUMBERS All numbers are complex and should be thought of in the form: a + bi Imaginary Part Real Part
COMPLEX NUMBERS a + bi Real Part Imaginary Part When b = 0, the number is a real number. Otherwise, the number is imaginary.
OPERATING ON COMPLEX NUMBERS Addition: Example: (3 + 5i) + ( - 2 + 3i) Subtraction: Example: (6 + 4i) - ( 3 + 6i)
OPERATING ON COMPLEX NUMBERS Multiplication: Example: (5 + 3i) • (2 + 7i) Example: (3 + 4i) • ( 3 - 4i)
2 + 3i = 2 - 3i CONJUGATES Multiplying a complex number by its conjugate always yields a nonnegative real number.
z z = a2 + b2 THEOREM: If z = a + bi
Writing the reciprocal of a complex number in standard form. Example:
Writing the quotient of complex numbers in standard form. Example:
Writing the quotient of complex numbers in standard form. Example:
POWERS OF i i1 = i i2 = - 1 i3 = - i i4 = 1 i5 = i and so on
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.
EXAMPLE Solve the following equations in the complex number system: x2 = 4 x2 = - 9
EXAMPLE • Solve the following equation in the complex number system: x2 - 4x + 8 = 0
DISCRIMINANT If b2 - 4ac > 0 Two unequal real sol’ns If b2 - 4ac = 0 One double real root If b2 - 4ac < 0 Two imaginary solutions
EXAMPLE: Without solving, determine the character of the solution of each equation in the complex number system: 3x2 + 4x + 5 = 0 2x2 + 4x + 1 = 0 9x2 - 6x + 1 = 0
