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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: 2.5.11.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 b2– 4ac is called the discriminant of a quadratic equation. If b2– 4ac > 0 (positive), the formula will give two real number solutions. If b2– 4ac = 0, there will be one real number solution, called a double root. If b2– 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 i 1 If r > 0, then the imaginary number r is defined as follows: r 1 r 1 r i r Example 1a Example 1b * -4x2 + 5x – 3 = 0 Example 1c * 6x2 – 3x + 1 = 0 Complex Numbers Example 1a and b* b. 2x + 3iy = -8 + 10i Operations with Complex Numbers c. (-10 – 6i) + (8 – i) Multiply a. (2 + i)(-5 – 3i) b. (6 – 4i)(5 – 4i) c. (2 – i)(-3 – 4i) 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 2 5i Rationalize the fraction: 2 3i 2 5i 2 3i Next step : FOIL 2 3i 2 3i (2)( 2) (2)(3i) (5i)( 2) (5i )(3i) (2)( 2) (2)(3i) (3i)( 2) (3i)(3i) 4 6i 10i 15i 2 2 4 6i 6i 9i 4 16i 15 49 11 16i 13 11 16 i 13 13 Example 1b 3 4i Rationalize the fraction: 2i Writing Questions Homework Pg. 320 #14-86 even