50 points answer. don't answer just for the points-

Assume z = x + iy, then find a complex number z satisfying the given equation. d. 2z8 – 2z4 + 1 = 0

kodGreya [7K]

Answer: complex equations has n number of solutions, been n the equation degree. In this case:

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i11,25°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i101,25°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i191,25°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i281,25°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i78,75°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i168,75°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i258,75°}$

$Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i348,75°}$

Step-by-step explanation:

I start with a variable substitution:

$Z^{4} = X$

Then:

$2X^{2}-2X+1=0$

Solving the quadratic equation:

$X_{1} =\frac{2+\sqrt{4-4*2*1} }{2*2} \\X_{2} =\frac{2-\sqrt{4-4*2*1} }{2*2}$

$X=\left \{ {{0,5+0,5i} \atop {0,5-0,5i}} \right.$

Replacing for the original variable:

$Z=\sqrt[4]{0,5+0,5i}$

or $Z=\sqrt[4]{0,5-0,5i}$

Remembering that complex numbers can be written as:

$Z=a+ib=|Z|e^{ic}$

Using this:

$Z=\left \{ {{{\frac{\sqrt{2}}{2} e^{i45°} } \atop {{\frac{\sqrt{2}}{2} e^{i-45°} }} \right.$

Solving for the modulus and the angle:

$Z=\left \{ {{\sqrt[4]{\frac{\sqrt{2}}{2} e^{i45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i45}} } \atop {\sqrt[4]{\frac{\sqrt{2}}{2} e^{i-45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i-45}} }} \right.$

The possible angle respond to:

$RAng_{12...n} =\frac{Ang +360*(i-1)}{n}$

Been "RAng" the resultant angle, "Ang" the original angle, "n" the degree of the root and "i" a value between 1 and "n"

In this case n=4 with 2 different angles: Ang = 45º and Ang = 315º

Obtaining 8 different angles, therefore 8 different solutions.

3 0

2 years ago

X/4 - y/3 =1 X and Y axis standard form

Semenov [28]

Answer:

Step-by-step explanation:

x/4 - y/3 = 1..multiply everything by the LCD....which is 12

3x - 4y = 12 <== standard form

=====================================

if it makes it easier for u, x/4 is the same as (1/4)x and y/3 is the same as (1/3)y

so ur problem can be written as : (1/4)x - (1/3)y = 1....multiply by 12

12(1/4)x - 12(1/3)y = 12(1)

(12/4)x - (12/3)y = 12.....reduce

3x - 4y = 12

5 0

3 years ago

PLEASE HELP MEEEEEEEE :)))

Kay [80]

Answer:

26

Step-by-step explanation:

Devaughn = 2x

Sydney = x

x + 2x = 78

3x = 78

x = 78 / 3

x = 26

Hope that helps!

8 0

3 years ago

Read 2 more answers

Enter the equation of the line in the form y=mx where m is the slope.Immersive Reader

kari74 [83]

Answer:

y = 1/2x

Step-by-step explanation:

The slope is the rise (up) over run (left or right depending on if the number is negative) between points. So the next point on the graph is up 5 and over 10. this simplified is 1/2. Then you put this into y=mx which is y=1/2x

4 0

3 years ago

How do i solve this

Sophie [7]

Step-by-step explanation:

You're given that ΔUVW is similar to ΔYZW. The tricky part is identifying which sides in ΔUVW correspond to which sides in ΔYZW.

The triangles share point W, so visually rotate one triangle around W until the two triangles align. That way, you can see that UW and ZW correspond to each other, and VW and YW correspond to each other.

Now you can find the scale:

VW / YW = 5/12

And you can write a proportion to find x:

UW / ZW = VW / YW

x / 64 = 5 / 12

x = 26 ⅔

5 0

3 years ago