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Transcript of Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic...
![Page 1: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/1.jpg)
Quadratic Equations and Complex Numbers
Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.
![Page 2: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/2.jpg)
![Page 3: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/3.jpg)
The Discriminant
![Page 4: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/4.jpg)
The Discriminant
![Page 5: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/5.jpg)
Example 1
![Page 6: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/6.jpg)
Example 1
![Page 7: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/7.jpg)
Example 1
![Page 8: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/8.jpg)
Example 1
![Page 9: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/9.jpg)
Try This
• Find the discriminant for each equation. Then, determine the number of real solutions.
01563 2 xx 0342 2 xx
![Page 10: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/10.jpg)
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
![Page 11: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/11.jpg)
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
![Page 12: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/12.jpg)
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
![Page 13: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/13.jpg)
![Page 14: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/14.jpg)
Imaginary Numbers
3313 i
222418 i
5359145 i
![Page 15: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/15.jpg)
Example 2
![Page 16: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/16.jpg)
Example 2
![Page 17: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/17.jpg)
Try This
• Use the quadratic formula to solve:
0354 2 xx
![Page 18: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/18.jpg)
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
![Page 19: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/19.jpg)
![Page 20: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/20.jpg)
Example 3
![Page 21: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/21.jpg)
Example 3
![Page 22: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/22.jpg)
Try This
• Find x and y such that 2x + 3iy = -8 + 10i
![Page 23: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/23.jpg)
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
![Page 24: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/24.jpg)
Example 4
![Page 25: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/25.jpg)
Example 4
![Page 26: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/26.jpg)
Additive Inverses
• Two complex numbers whose real parts are opposites and whose imaginary parts are opposites are called additive inverses.
0)34()34( ii
![Page 27: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/27.jpg)
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
![Page 28: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/28.jpg)
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
![Page 29: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/29.jpg)
Example 5
![Page 30: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/30.jpg)
Example 5
![Page 31: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/31.jpg)
Try This
• Multiply )45)(46( ii
![Page 32: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/32.jpg)
Try This
• Multiply )45)(46( ii
ii
iii
4414)1(164430
16202430 2
![Page 33: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/33.jpg)
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.
![Page 34: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/34.jpg)
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
![Page 35: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/35.jpg)
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
![Page 36: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/36.jpg)
Example 6
• Simplify . Write your answer in standard form. i
i
32
52
![Page 37: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/37.jpg)
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
![Page 38: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/38.jpg)
Example 6
• Simplify . Write your answer in standard form. i
i
2
43
![Page 39: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/39.jpg)
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
![Page 40: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/40.jpg)
![Page 41: Quadratic Equations and Complex Numbers Objective: Classify and find all roots of a quadratic equation. Perform operations on complex numbers.](https://reader033.fdocuments.us/reader033/viewer/2022061305/5513fae1550346ec488b4602/html5/thumbnails/41.jpg)
Homework
• Page 320• 24-66 multiples of 3