Download Reading Assignment: Chapter 5, Pages 314 – 319

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