Download 5.6 – Quadratic Equations and Complex Numbers

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.
Discriminant
The expression b2– 4ac is called the discriminant
of a quadratic equation.
 If b2– 4ac > 0 (positive), the formula will give two
different answers.
 If b2– 4ac = 0, there will be one answer, 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
If r > 0, then the imaginary number
follows:
 r = 1  r = i r
Example 1a
 r is defined as
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)
(-10 + 8) + (-6i – i)
-2 – 7i
Multiply (Example 2)
b. (6 – 4i)(5 – 4i)  FOIL
c. (2 – i)(-3 – 4i)  FOIL
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.
Example 1a
2  5i
2  3i
2  5i 2  3i

Next step : FOIL
2  3i 2  3i
Example 1b
3  4i
2i
3  4i 2  i

2i 2i
Writing Questions