5.6 – Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation. Graph and perform operations on complex numbers. Standard: C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

The Solutions to a Quadratic Equation can referred to as ANY of the following: x – intercepts Solutions Roots Zeroes

Discriminant The expression b 2 – 4ac is called the discriminant of a quadratic equation. If b 2 – 4ac > 0 (positive), the formula will give two real number solutions. If b 2 – 4ac = 0, there will be one real number solution, called a double root. If b 2 – 4ac < 0 (negative), the formula gives no real solutions

Ex 1. Find the discriminant for each equation. Then determine the number of real solutions for each equation by using the discriminant.

Imaginary Numbers If r > 0, then the imaginary number is defined as follows: Example 1a

Example 1b * -4x 2 + 5x – 3 = 0

Example 1c * 6x 2 – 3x + 1 = 0

Complex Numbers

Example 1a and b* b. 2x + 3iy = i

Operations with Complex Numbers c. (-10 – 6i) + (8 – i)

Multiply c. (2 – i)(-3 – 4i) b. (6 – 4i)(5 – 4i) a. (2 + i)(-5 – 3i)

Conjugate of a Complex Number The conjugate of a complex number a + bi is a – bi. To simplify a quotient with an imaginary number in the denominator, multiply by a fraction equal to 1, using the conjugate of the denominator. This process is called rationalizing the denominator.

4+3i 5 - 4i -7+ 6i -9 - i

Example 1a Rationalize the fraction:

Example 1b Rationalize the fraction:

Writing Questions

Homework Pg. 320 #14-86 even