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ALGEBRA 2 LECTURE Q – 4: Quadratic Equations and Complex Numbers Reading Assignment: Chapter 5, Pages 314 – 319 THE DISCRIMINANT When you apply the quadratic formula to any quadratic equation, you will find that the value of b2 – 4ac is either positive, negative, or 0. The expression b2 – 4ac is called the discriminant of a quadratic equation. If b2 – 4ac 0 the quadratic has two distinct solutions, roots, or zeroes If b2 – 4ac = 0 the quadratic has a 1 real solution – a double root (one zero that occurs twice) If b2 – 4ac 0 the quadratic has no real solutions EXAMPLE 1: Find the discriminant and determine the number of real solutions for each equation A. 2x2 + 4x + 1 = 0 B. 2x2 + 4x + 2 = 0 C. 2x2 + 4x + 3 = 0 TRY THIS Page 315: Identify the number of real solutions to – 3x2 – 6x + 15 = 0 IMAGINARY NUMBERS If r 0, then the imaginary number √−𝑟 is defined as √−𝑟 = √−1 √𝑟 = i√𝑟 EXAMPLE 2: Use the quadratic formula to solve 3x2 – 7x + 5 = 0 TRY THIS Page 316: Use the quadratic Formula to solve –4x2 + 5x – 3 = 0 ALGEBRA 2 LECTURE Q – 4: Quadratic Equations and Complex Numbers COMPLEX NUMBERS A complex number is any number that can be written as a + bi where a and b are real numbers and i = √−1 a is called the real part and b is called the imaginary part EXAMPLE 3: Find x and y such that 7x – 2iy = 14 + 6i EXAMPLE 4: Find each sum or difference A. (–3 + 5i) + (7 – 6i) B. (–3 – 8i) – (–2 – 9i) EXAMPLE 5: Multiply (2 + i) ( –5 – 3i) TRY THIS Page 316: Find x and y such that 2x + 3iy = –8 + 10i TRY THIS Page 317: Multiply (6 – 4i) ( 5 – 4i) CONJUGATE OF A COMPLEX NUMBER The conjugate of a complex number a + bi is a – bi. You often need to use the conjugate of a complex number to simplify a fraction containing complex numbers. Rationalizing the denominator: simplify a quotient with an imaginary number in the denominator by multiplying by a fraction equal to 1 using the conjugate of the denominator. EXAMPLE 6: Simplify 2+5𝑖 2−3𝑖 TRY THIS Page 318: Simplify 3−4𝑖 2+𝑖 ALGEBRA 2 LECTURE Q – 4: Quadratic Equations and Complex Numbers THE COMPLEX PLANE Complex numbers are graphed in the complex plane. The horizontal axis of the complex plane is the real axis. The vertical axis of the complex plane is the imaginary axis. To graph the complex number a + bi, plot the point (a, b). The absolute value of a complex number is its distance from the origin in the complex plane. By the Pythagrean Theorem: a + bi= √𝑎2 + 𝑏 2 . EXAMPLE 7: Evaluate –2 – 3i. Sketch a diagram that shows –2 – 3i and –2 – 3i TRY THIS Page 319: Evaluate –3 + 5i. Sketch a diagram that shows –3 + 5i and –3 + 5i ALGEBRA 2 LECTURE Q – 4: Quadratic Equations and Complex Numbers HW Q – 4 Page 320 #19, 21, 23, 33, 35, 37, 43, 45, 53, 55, 65, 73, 77, 79, 93