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Quadratic Equations and
Complex Numbers
Objective: Classify and find all roots
of a quadratic equation. Perform
operations on complex numbers.
The Discriminant
The Discriminant
Example 1
Example 1
Example 1
Example 1
Try This
• Find the discriminant for each equation. Then,
determine the number of real solutions.
 3x  6 x  15  0
2
2x  4x  3  0
2
Try This
• Find the discriminant for each equation. Then,
determine the number of real solutions.
 3x  6 x  15  0
2
(6) 2  4(3)(15)  216
2 real roots
2x  4x  3  0
2
Try This
• Find the discriminant for each equation. Then,
determine the number of real solutions.
 3x  6 x  15  0
2
2x  4x  3  0
2
(6) 2  4(3)(15)  216
(4) 2  4(2)(3)  8
2 real roots
0 real roots
Imaginary Numbers
• If the discriminant is negative, that means when
using the quadratic formula, you will have a negative
number under a square root. This is what we call an
imaginary number and is defined as:
i  1
i  1
2
Imaginary Numbers
 3  1 3  i 3
 8   1 4 2  2i 2
 45   1 9 5  3i 5
Example 2
Example 2
Try This
• Use the quadratic formula to solve:
 4 x 2  5x  3  0
Try This
• Use the quadratic formula to solve:
 4 x 2  5x  3  0
 5  (5)  4(4)( 3)
2(4)
2
 5  25  48  5   23 5 i 23

 
8
8
8 8
Example 3
Example 3
Try This
• Find x and y such that
2x + 3iy = -8 + 10i
Try This
• Find x and y such that
2x + 3iy = -8 + 10i
2 x  8
x  4
real part
3iy  10i
3 y  10
y  103
imaginary part
Example 4
Example 4
Additive Inverses
• Two complex numbers whose real parts are opposites
and whose imaginary parts are opposites are called
additive inverses.
(4  3i )  (4  3i )  0
Additive Inverses
• Two complex numbers whose real parts are opposites
and whose imaginary parts are opposites are called
additive inverses.
(4  3i )  (4  3i )  0
• What is the additive inverse of 2i – 12?
Additive Inverses
• Two complex numbers whose real parts are opposites
and whose imaginary parts are opposites are called
additive inverses.
(4  3i )  (4  3i )  0
• What is the additive inverse of 2i – 12?
-2i + 12
Example 5
Example 5
Try This
• Multiply
(6  4i )(5  4i )
Try This
• Multiply
(6  4i )(5  4i )
30  24i  20i  16i 2
30  44i  16(1)  14  44i
Conjugate of a Complex Number
• In order to simplify a fraction containing complex
numbers, you often need to use the conjugate of a
complex number. For example, the conjugate of
2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.
Conjugate of a Complex Number
• In order to simplify a fraction containing complex
numbers, you often need to use the conjugate of a
complex number. For example, the conjugate of
2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.
________
• The conjugate of a  bi is denoted a  bi .
Conjugate of a Complex Number
• In order to simplify a fraction containing complex
numbers, you often need to use the conjugate of a
complex number. For example, the conjugate of
2 + 5i is 2 – 5i and the conjugate of 1 – 3i is 1 + 3i.
________
• The conjugate of a  bi is denoted a  bi .
• To simplify a quotient with an imaginary number,
multiply by 1 using the conjugate of the denominator.
Example 6
• Simplify
form.
2  5i
. Write your answer in standard
2  3i
Example 6
• Simplify
form.
2  5i
. Write your answer in standard
2  3i
• Multiply the top and bottom by 2 + 3i.
2  5i 2  3i 4  6i  10i  15i 2
11 16i


 
2
2  3i 2  3i
4  6i  6i  9i
13 13
Example 6
• Simplify
form.
3  4i
. Write your answer in standard
2i
Example 6
• Simplify
form.
3  4i
. Write your answer in standard
2i
• Multiply the top and bottom by 2 – i.
3  4i 2  i 6  3i  8i  4i 2
2 11i


 
2
2  i 2  i 4  2i  2i  i
5 5
Homework
• Page 320
• 24-66 multiples of 3