5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex Numbers Numbers Objectives: •Graph and perform operations on complex numbers
Jan 01, 2016
5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
Objectives: •Graph and perform operations on complex numbers
Imaginary 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.
3 + 4i
real part
imaginary part
3 4i
Example 1Find x and y such that -3x + 4iy = 21 – 16i.
Real parts Imaginary parts-3x = 21
x = -7
4y = -16y = -
4
x = -7 and y = -4
Example 2Find each sum or difference.
a) (-10 – 6i) + (8 – i) = (-10 +
8) = -2 – 7i
b) (-9 + 2i) – (3 – 4i)= (-9 –
3)= -12 + 6i
+ (2i + 4i)
+ (-6i – i)
Example 3Multiply.
(2 – i)(-3 – 4i)
= -6- 8i + 3i
+ 4i2
= -6- 5i + 4(-1)
= -10 – 5i
Conjugate of a Complex Number
The conjugate of a complex number a + bi is a – bi.The conjugate of a + bi is denoted a + bi.
Example 4
multiply by 1, using the conjugate of the denominator
3 2iSimplif y . Write your answer in standard f orm.
4 i
3 2i4 i
4 i4 i
=(3 – 2i)(-4 + i)
(-4 – i)
(-4 - i)
=-12
16
- 3i
+ 4i
+ 8i+ 2i2- 4i - i2
=-12
16
+ 5i+ 2(-1)- (-1)
=-14
17
+ 5i
Practice3 4i
Simplif y . Write your answer in standard f orm.2 i
Warm-UpPerform the indicated operations, and simplify.
5 minutes
1) (-4 + 2i) + (6 – 3i)
2) (2 + 5i) – (5 + 3i)
3) (7 + 7i) – (-6 – 2i)
4) (2 i 5)( 1 i 5)
Warm-UpUse the quadratic formula to solve each equation.
6 minutes
1) x2 + 12x + 35 = 0
2) x2 + 81 = 18x
3) x2 + 4x – 9 = 0 4) 2x2 = 5x + 9
5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
Objectives: •Classify and find all roots of a quadratic equation
Solutions of a Quadratic Equation
If b2 – 4ac > 0, then the quadratic equation has 2 distinct real solutions.
Let ax2 + bx + c = 0, where a = 0.
If b2 – 4ac = 0, then the quadratic equation has 1 real solutions.
If b2 – 4ac < 0, then the quadratic equation has 0 real solutions.
The expression b2 – 4ac is called the discriminant.
Example 1Find the discriminant for each equation. Then determine the number of real solutions.
a) 3x2 – 6x + 4 = 0 b2 – 4ac
= (-6)2 – 4(3)(4) =
36 – 48 =
-12
no real solutions
b) 3x2 – 6x + 3 = 0 b2 – 4ac
= (-6)2 – 4(3)(3) =
36 – 36 =
0
one real solution
c) 3x2 – 6x + 2 = 0 b2 – 4ac
= (-6)2 – 4(3)(2) =
36 – 24 =
12
two real solutions
PracticeIdentify the number of real solutions:
1) -3x2 – 6x + 15 = 0
Imaginary NumbersThe imaginary unit is defined as and i2 = -1.
i 1
If r > 0, then the imaginary number is defined as follows:
r
r 1 r i r
10 1 10 i 10
Example 2Solve 6x2 – 3x + 1 = 0. 2b b 4ac
x2a
23 3 4(6)(1)x
2(6)
3 9 24x
12
3 15x
12
3 i 15x
12
PracticeSolve -4x2 + 5x – 3 = 0.
Warm-UpFind the discriminant, and determine the number of real solutions. Then solve.
5 minutes
1) x2 – 7x = -10 2) 5x2 + 4x = -5
5.6.3 Quadratic Equations and Complex 5.6.3 Quadratic Equations and Complex NumbersNumbers
5.6.3 Quadratic Equations and Complex 5.6.3 Quadratic Equations and Complex NumbersNumbers
Objectives: •Graph and perform operations on complex numbers
The Complex PlaneIn the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis.
-4 -2
2
42
4
-4
-2
real axis
imaginary axis