1. Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex Numbers Solve Quadratic Equations in the Complex Number System

2. The equation x 2 = - 1 has no real number solution. To remedy this situation, we define a new number that solves this equation, called the imaginary unit, which is not a real number. The solution to x 2 = - 1 is the imaginary unit I where i 2 = - 1, or

3. Complex numbers are numbers of the form a + bi, where a and b are real numbers. The real number a is called the real part of the number a + bi; the real number b is called the imaginary part of a + bi. Equality of Complex Numbers a + bi = c + di if and only if a = c and b = d In other words, complex numbers are equal if and only if there real and imaginary parts are equal.

4. Addition with Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i Example: (2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i = 1 + 10i

5. Subtraction with Complex Numbers (a + bi) - (c + di) = (a - c) + (b - d)i Example (3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i = 2 + 3i

6. Example: Multiply using the distributive property Multiplication with Complex Numbers

7. If z = a + bi is a complex number, then its conjugate, denoted by, is defined as Theorem The product of a complex number and its conjugate is a nonnegative real number. Thus, if z = a + bi, then

8. Division with Complex Numbers To divide by a complex number, multiply the dividend (numerator) and divisor (denominator) by the conjugate of the divisor. Example:

9. In the complex number system, the solutions of the quadratic equation where a, b, and c are real numbers and a 0, are given by the formula Since we now have a way of evaluating the square root of a negative number, there are now no restrictions placed on the quadratic formula.

10. Find all solutions to the equation real or complex.