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3 years ago

Add or subtract. Write the answer in standard form. 7−(3+4i)+6i

Mathematics

1 answer:

Alisiya [41]3 years ago

3 0

Answer:

4 + 2i

Step-by-step explanation:

First distribute;

7 - 3 - 4i + 6i

= 4 + 2i

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Find the solution of the given initial value problem: y''- y = 0, y(0) = 2, y'(0) = -1/2

igor_vitrenko [27]

Answer: The required solution of the given IVP is

$y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.$

Step-by-step explanation: We are given to find the solution of the following initial value problem :

$y^{\prime\prime}-y=0,~~~y(0)=2,~~y^\prime(0)=-\dfrac{1}{2}.$

Let $y=e^{mx}$ be an auxiliary solution of the given differential equation.

Then, we have

$y^\prime=me^{mx},~~~~~y^{\prime\prime}=m^2e^{mx}.$

Substituting these values in the given differential equation, we have

$m^2e^{mx}-e^{mx}=0\\\\\Rightarrow (m^2-1)e^{mx}=0\\\\\Rightarrow m^2-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mx}\neq0]\\\\\Rightarrow m^2=1\\\\\Rightarrow m=\pm1.$

So, the general solution of the given equation is

$y(x)=Ae^x+Be^{-x},$ where A and B are constants.

This gives, after differentiating with respect to x that

$y^\prime(x)=Ae^x-Be^{-x}.$

The given conditions implies that

$y(0)=2\\\\\Rightarrow A+B=2~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

and

$y^\prime(0)=-\dfrac{1}{2}\\\\\\\Rightarrow A-B=-\dfrac{1}{2}~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

Adding equations (i) and (ii), we get

$2A=2-\dfrac{1}{2}\\\\\\\Rightarrow 2A=\dfrac{3}{2}\\\\\\\Rightarrow A=\dfrac{3}{4}.$

From equation (i), we get

$\dfrac{3}{4}+B=2\\\\\\\Rightarrow B=2-\dfrac{3}{4}\\\\\\\Rightarrow B=\dfrac{5}{4}.$

Substituting the values of A and B in the general solution, we get

$y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.$

Thus, the required solution of the given IVP is

$y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.$

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3 years ago

Write an equation in point slope form for a line that includes the origin and (9,-3)

aivan3 [116]

So we are given two points that the line crosses, the origin and (9, -3), we can calculate the slope m of the line with these data, dividing the y segment by the x segment:
m = (-3 - 0)/(9 - 0) = -3/9
m = -1/3
then we can use the point slope line equation to find the line equation, lets use the point origin (0,0) to do so:
y - y1 = m(x - x1), where x1, y1 are the coordinates of a point that the line crosses:
y - 0 = (-1/3)(x - 0)
y = (-1/3)x
so this is the equation of the line, slope -1/3 and y intercept 0

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3 years ago

Evaluate p(-2)=x^3-4x-5

Pani-rosa [81]

Step-by-step explanation:

$p(x) = {x}^{3} - 4x - 5 \\ p( - 2) = {( - 2)}^{3} - 4( - 2) - 5 \\ = - 8 + 8 - 5 \\ = - 5$