# 3_5 - Example 1 Example 2 2 Solving Quadratic Equations...

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Section 3.5: Complex Numbers; Quadratic Equations with NEGATIVE Discriminants Instructor: Ms. Hoa Nguyen 1 Complex Numbers The complex number i is the solution of the equation x 2 = - 1, i.e., i 2 = - 1. We write i = - 1. The complex number in the standard form is: a + bi where a, b are real numbers. a is the real part and b is the imaginary part of the complex number. A REAL number is a complex number with b = 0. So, the set of real numbers is a subset of the set of complex numbers. Algebraic operations: Equality: a + bi = c + di a = c and b = d . Addition: ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i . Subtraction: ( a + bi ) - ( c + di ) = ( a - c ) + ( b - d ) i . Multiplication: ( a + bi )( c + di ) = ( ac - bd ) + ( ad + bc ) i . Conjugate: If z = a + bi , then the conjugate of z is ¯ z = a - bi . Remark z + ¯ z = 2 a z - ¯ z = 2 bi z ¯ z = a 2 + b 2 = | z | 2 ¯ ¯ z = z z + w = ¯ z + ¯ w z · w = ¯ z · ¯ w A reciprocal of a nonzero complex number z = a + bi is 1 z = ¯ z z ¯ z = a a 2 + b 2 - b a 2 + b 2 i . Division: If z = a + bi and w = c + di , then z w = z · ¯ w w ¯ w = z · ¯ w c 2 + d 2 . 1

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Powers of i : i 1 = i i 2 = - 1 i 3 = i 2 i = - i i 4 = ( i 2 ) 2 = ( - 1) 2 = 1 i 5 = i 4 i = i The powers of i repeat with EVERY 4th power.
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Unformatted text preview: Example 1 Example 2 2 Solving Quadratic Equations with NEGATIVE Dis-criminants Quadratic Formula : In the complex number system, the solutions of a quadratic equation ax 2 + bx + c = 0, where a 6 = 0 , a, b, c are real numbers can be computed by the quadratic formula: x =-b ± √ b 2-4 ac 2 a (1) Let D = b 2-4 ac be the discriminant : 2 • D > 0: 2 REAL solutions (2 x-intercepts) • D = 0: 1 REAL REPEATED/ DOUBLED solution (1 x-intercept) • D < 0: 2 COMPLEX (NOT REAL) solutions (0 x-intercept). In this case, the solutions are CONJUGATES of each other . It means that if a quadratic function of real coeﬃcients has a complex number z as a zero, then the complex conjugate ¯ z is also a zero. Remark : √-1 = i ⇒ √-16 = √-1 √ 16 = 4 i . Example 3 3...
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3_5 - Example 1 Example 2 2 Solving Quadratic Equations...

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