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3 years ago

What is the quotient of the complex number 4-3i divided by its conjugate?

Mathematics

1 answer:

almond37 [142]3 years ago

3 0

Answer: C. $\frac{7}{25}-\frac{24}{25}i$

Step-by-step explanation:

1. You have the following division:

$\frac{4-3i}{4+3i}$ (As you can see, to find the conjugate of 4-3i you must change the sign between the terms).

2. To solve this division, you must multiply the numerator and the denominator by the conjugate of the denominator, as following:

$=\frac{(4-3i)}{(4+3i)}\frac{(4-3i)}{(4-3i)}=\frac{16-12i-12i+9i^{2}}{16-9i^{2}}$

3. Keeping on mind that $i^{2}=-1$ , you have:

$=\frac{16-12i-12i+9(-1)}{16-9(-1)}$

4. Simplifying:

$=\frac{7-24i}{25}=\frac{7}{25}-\frac{24}{25}i$

5. The result is:

$\frac{7}{25}-\frac{24}{25}i$

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OlgaM077 [116]

The answer is C.

This is because 7 must be greater than 12-g.

G cannot equal 5 and below because 12-5=7.

Therefore C is the only answer that works.

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2 years ago

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for every 30 minutes of TV air time there are 8 minutes of commercial. If 90 minutes of TV or are now how many how many minutes

Anna35 [415]

Let m = minutes of commercial

30/90 = 8/m

Solve for m.

90(8) = 30m

720 = 30m

720/30 = m

24 = m

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2 years ago

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Consider the following. (A computer algebra system is recommended.) y'' + 3y' = 2t4 + t2e−3t + sin 3t (a) Determine a suitable f

drek231 [11]

First look for the fundamental solutions by solving the homogeneous version of the ODE:

$y''+3y'=0$

The characteristic equation is

$r^2+3r=r(r+3)=0$

with roots $r=0$ and $r=-3$ , giving the two solutions $C_1$ and $C_2e^{-3t}$ .

For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.

$y''+3y'=2t^4$

Assume the ansatz solution,

${y_p}=at^5+bt^4+ct^3+dt^2+et$

$\implies {y_p}'=5at^4+4bt^3+3ct^2+2dt+e$

$\implies {y_p}''=20at^3+12bt^2+6ct+2d$

(You could include a constant term f here, but it would get absorbed by the first solution $C_1$ anyway.)

Substitute these into the ODE:

$(20at^3+12bt^2+6ct+2d)+3(5at^4+4bt^3+3ct^2+2dt+e)=2t^4$

$15at^4+(20a+12b)t^3+(12b+9c)t^2+(6c+6d)t+(2d+e)=2t^4$

$\implies\begin{cases}15a=2\\20a+12b=0\\12b+9c=0\\6c+6d=0\\2d+e=0\end{cases}\implies a=\dfrac2{15},b=-\dfrac29,c=\dfrac8{27},d=-\dfrac8{27},e=\dfrac{16}{81}$

$y''+3y'=t^2e^{-3t}$

$e^{-3t}$ is already accounted for, so assume an ansatz of the form

$y_p=(at^3+bt^2+ct)e^{-3t}$

$\implies {y_p}'=(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}$

$\implies {y_p}''=(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}$

Substitute into the ODE:

$(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}+3(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}=t^2e^{-3t}$

$9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c-9at^3+(9a-9b)t^2+(6b-9c)t+3c=t^2$

$-9at^2+(6a-6b)t+2b-3c=t^2$

$\implies\begin{cases}-9a=1\\6a-6b=0\\2b-3c=0\end{cases}\implies a=-\dfrac19,b=-\dfrac19,c=-\dfrac2{27}$

$y''+3y'=\sin(3t)$

Assume an ansatz solution

$y_p=a\sin(3t)+b\cos(3t)$

$\implies {y_p}'=3a\cos(3t)-3b\sin(3t)$

$\implies {y_p}''=-9a\sin(3t)-9b\cos(3t)$

Substitute into the ODE:

$(-9a\sin(3t)-9b\cos(3t))+3(3a\cos(3t)-3b\sin(3t))=\sin(3t)$

$(-9a-9b)\sin(3t)+(9a-9b)\cos(3t)=\sin(3t)$

$\implies\begin{cases}-9a-9b=1\\9a-9b=0\end{cases}\implies a=-\dfrac1{18},b=-\dfrac1{18}$

So, the general solution of the original ODE is

$y(t)=\dfrac{54t^5 - 90t^4 + 120t^3 - 120t^2 + 80t}{405}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\dfrac{3t^3+3t^2+2t}{27}e^{-3t}-\dfrac{\sin(3t)+\cos(3t)}{18}$

3 0

3 years ago

URGENT!! NEED THIS IN TWO HOURS. DUMB ANSWERS WILL BE REPORTED ALTHOUGH I UNDERSTAND IF YOU MAKE A SIMPLE MISTAKE IN YOUR CALCUL

Andre45 [30]

Yo sup??

The given points are (1,1),(1,3) and (9,2)

equation of circle is

8x²+8y²+ax+by=0

let us plug in (1,1) in the equation we get

8+8+a+b=0

a+b=-16

let us now plug in (1,3) in the equation

8+8*3²+a+3b=0

a+3b=-80

let us. now solve the two equation in terms of a and b

we get

2b=--64

b=-32

and

a=-16+32

=16

therefore the equation is

8x²+8y²+16x-32y=0

Hope this helps

6 0

2 years ago

Andrew borrowed $700 from the bank to start a pressure washing business. The bank charged a simple interest rate of 7% per year.

Ede4ka [16]

Answer:

$147

Step-by-step explanation:

Given:

Loan amount = $700
Interest rate = 7% simple
Time = 3 years

Interest amount:

$700*3*0.07 = $147

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2 years ago

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