quadratic equations and complex numbers objective: classify and find all roots of a quadratic...
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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.
01563 2 xx 0342 2 xx
Try This
• Find the discriminant for each equation. Then, determine the number of real solutions.
2 real roots
01563 2 xx 0342 2 xx
216)15)(3(4)6( 2
Try This
• Find the discriminant for each equation. Then, determine the number of real solutions.
2 real roots 0 real roots
01563 2 xx 0342 2 xx
216)15)(3(4)6( 2 8)3)(2(4)4( 2
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:
1i
12 i
Imaginary Numbers
3313 i
222418 i
5359145 i
Example 2
Example 2
Try This
• Use the quadratic formula to solve:
0354 2 xx
Try This
• Use the quadratic formula to solve:
0354 2 xx
)4(2
)3)(4(4)5(5 2
8
23
8
5
8
235
8
48255
i
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
real part imaginary part
4
82
x
x
310
103
103
y
y
iiy
Example 4
Example 4
Additive Inverses
• Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.
0)34()34( ii
Additive Inverses
• Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.
• What is the additive inverse of 2 – 12i?
0)34()34( ii
Additive Inverses
• Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.
• What is the additive inverse of 2 – 12i? -2 + 12i
0)34()34( ii
Example 5
Example 5
Try This
• Multiply )45)(46( ii
Try This
• Multiply )45)(46( ii
ii
iii
4414)1(164430
16202430 2
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 is denoted .bia ________
bia
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 is denoted .
• To simplify a quotient with an imaginary number, multiply by 1 using the conjugate of the denominator.
bia ________
bia
Example 6
• Simplify . Write your answer in standard form. i
i
32
52
Example 6
• Simplify . Write your answer in standard form.
• Multiply the top and bottom by 2 + 3i.
i
i
32
52
13
16
13
11
9664
151064
32
32
32
522
2 i
iii
iii
i
i
i
i
Example 6
• Simplify . Write your answer in standard form. i
i
2
43
Example 6
• Simplify . Write your answer in standard form.
• Multiply the top and bottom by 2 – i.
i
i
2
43
5
11
5
2
224
4836
2
2
2
432
2 i
iii
iii
i
i
i
i
Homework
• Page 320• 24-66 multiples of 3