![Page 1: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/1.jpg)
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
![Page 2: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/2.jpg)
5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
Objectives: •Classify and find all roots of a quadratic equation
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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.
![Page 4: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/4.jpg)
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
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PracticeIdentify the number of real solutions:
1) -3x2 – 6x + 15 = 0
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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
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Example 2Solve 6x2 – 3x + 1 = 0. 2b b 4acx 2a
23 3 4(6)(1)x 2(6)
3 9 24x 12
3 15x 12
3 i 15x 12
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PracticeSolve -4x2 + 5x – 3 = 0.
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Homework
p.320 #15-25 odds,29,33,37,43
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Warm-UpFind the discriminant, and determine the number of real solutions. Then solve.
5 minutes
1) x2 – 7x = -10 2) 5x2 + 4x = -5
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5.6 Quadratic Equations and Complex 5.6 Quadratic Equations and Complex NumbersNumbers
Objectives: •Graph and perform operations on complex numbers
![Page 12: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/12.jpg)
Imaginary NumbersA 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
![Page 13: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/13.jpg)
Example 1Find x and y such that -3x + 4iy = 21 – 16i.
Real parts Imaginary parts-3x = 21
x = -74y = -16y = -
4x = -7 and y = -4
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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)
![Page 15: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/15.jpg)
Example 3Multiply.
(2 – i)(-3 – 4i)= -6- 8i +
3i+ 4i2
= -6- 5i + 4(-1)= -10 – 5i
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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.
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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)
= -1216
- 3i+ 4i
+ 8i+ 2i2- 4i - i2
= -1216
+ 5i+ 2(-1)- (-1) = -14
17
+ 5i
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Practice3 4iSimplif y . Write your answer in standard f orm.2 i
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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)
![Page 20: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/20.jpg)
5.6.3 Quadratic Equations and Complex 5.6.3 Quadratic Equations and Complex NumbersNumbers
Objectives: •Graph and perform operations on complex numbers
![Page 21: Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9](https://reader035.vdocuments.mx/reader035/viewer/2022062413/5a4d1b5b7f8b9ab0599aaf66/html5/thumbnails/21.jpg)
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
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Homework
worksheet 5.6 “B”