X ^2 + 2x + y’ + 6y = 15

2 Answers:

z = 1 + i   and   z = -1 - i

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Explanation:

We want z to be a complex number in the form z = a+bi, where a,b are real numbers and [tex]i = \sqrt{-1}[/tex] is imaginary.

Let's plug that into the equation your teacher gave you

[tex]z^2 = 2i\\\\(a+bi)^2 = 2i\\\\(a+bi)(a+bi) = 2i\\\\a(a+bi)+bi(a+bi) = 2i\\\\a^2+abi+abi+b^2*i^2 = 2i\\\\a^2+2abi+b^2*(-1) = 2i\\\\a^2+2abi-b^2 = 2i\\\\(a^2-b^2)+2abi = 0+2i\\\\[/tex]

You could use the FOIL rule to take a shortcut. I'm deciding to be a bit more wordy to show a further breakdown how everything is multiplying out.

Notice that the real part a^2-b^2 must be 0 so that it matches the real part on the right hand side.

a^2-b^2 = 0

(a-b)(a+b) = 0 .... difference of squares rule

a-b = 0 or a+b = 0

a = b or a = -b

So whatever solution z = a+bi is, it must have either a = b or a = -b.

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If a = b, then the 2abi portion on the left side turns into 2a^2*i

Set this equal to 2i on the right hand side and isolate 'a'

[tex]2a^2*i = 2i\\\\2a^2 = 2\\\\a^2 = 1\\\\a = 1 \text{ or } a = -1\\\\[/tex]

So a = 1 leads to b = 1

Or a = -1 leads to b = -1

Two complex solutions so far are:   z = 1 + i   and   z = -1 - i  based on those two cases above.

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Now consider the case that a = -b

We'll effectively have the same steps as the previous section, but the equation to solve now is [tex]-2a^2*i = 2i\\\\[/tex]

The only difference is that negative is out front. You should find that it leads to a^2 = -1, but this has no solutions because we stated earlier that a,b were real numbers.

So if a = -b, then it concludes with a,b being nonreal numbers. Ultimately we rule out the case that a = -b is possible.

Put another way, note how -2a^2 is always negative which clashes with the idea that the right hand side is positive (ignore the 'i' portions). This contradiction means that no real values of 'a' will make the equation [tex]-2a^2*i = 2i\\\\[/tex] to be true.

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To wrap things up, we only have two solutions and they are  

z = 1 + i   and   z = -1 - i

You can use a tool like WolframAlpha to confirm this.

Answer:

Hello,

Answer A (-4,0) and (0,0)

Step-by-step explanation:

[tex]\left\{\begin{array}{ccc}y&=&x^2+4x\\y+x^2&=&-4x\\\end{array} \right.\\\\\\\left\{\begin{array}{ccc}y&=&x^2+4x\\y&=&-x^2-4x\\\end{array} \right.\\\\\\\left\{\begin{array}{ccc}y&=&x^2+4x\\x^2+4x&=&-x^2-4x\\\end{array} \right.\\\\\\\left\{\begin{array}{ccc}2*x^2+8*x&=&0\\y&=&x^2+4x\\\end{array} \right.\\\\\\\left\{\begin{array}{ccc}x(x+4)&=&0\\y&=&x^2+4x\\\end{array} \right.\\\\\\[/tex]

[tex]\left\{\begin{array}{ccc}x&=&0 \\y&=&0\\\end{array} \right. \ or\ \left\{\begin{array}{ccc}x&=&-4 \\y&=&0\\\end{array} \right.[/tex]

Answer:

The total number of ways is 252.

Step-by-step explanation:

Number of girls = 18

number of boys = 14

Commitee of one girl and a boy

(18 C 1)(14 C 1)

= 252

Answer:

0.000000073

Step-by-step explanation:

Given number is,

7.3E - 8

In scientific notation number will be,

7.3 × 10⁻⁸

In standard form the number will be,

0.000000073

Answer:

a) 0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

b) The expected number of at-bats until the next home run is 6.25.

Step-by-step explanation:

For each at bat, there are two possible outcomes. Either it is a home run, or it is not. The probability of an at bat resulting in a home run is independent of any other at-bat, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

The probability that Barry Bonds hits a home run on any given at-bat is 0.16

This means that [tex]p = 0.16[/tex]

Part A: What is the probability that the next home run will be on his fifth at-bat?

0 on his next 4(P(X = 0) when n = 4)

Home run on his 5th at-bat, with 0.16 probability. So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{4,0}.(0.16)^{0}.(0.84)^{4} = 0.49787136 [/tex]

0.49787136 *0.16 = 0.0797.

0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

Part B: What is the expected number of at-bats until the next home run?

The expected number of trials for n successes is given by:

[tex]E = \frac{n}{p}[/tex]

In this question, [tex]n = 1, p = 0.16[/tex]. So

[tex]E = \frac{1}{0.16} = 6.25[/tex]

The expected number of at-bats until the next home run is 6.25.

Answer:

[tex]\frac{2}{5}^{-3}[/tex]×[tex]x=\frac{1}{2}^{4}[/tex]

[tex]x=\frac{2}{5} ^{3} \\[/tex]×[tex]\frac{1}{2}^{4}[/tex]

(negative in the exponent means reciprocal of the fraction)

x= [tex]\frac{1}{250}[/tex]

Brainliest please

Answer:

B. 3.39 m

Step-by-step explanation:

r² = A/π

= 36/3.14

= 11.465

r = √11.465 = 3.39

9514 1404 393

Answer:

  B. a = plus-or-minus 4

Step-by-step explanation:

  3a² -21 = 27 . . . . . . . given

  3a² = 48 . . . . . . . . . . add 21 to both sides (desired form)

  a² = 16 . . . . . . . . . . .  divide both sides by 3

  a = ±4 . . . . . . . take the square root

It doesn't change because to add fractions, you need a common denominator. To find it, they multiplied 1/3 by 2 to make 2/6, to add to the 3/6.

Answer:

horizontal expansion factor of 2

2^x =2(2)=4

2^4=2×2×2×2= 16

Answer:

0.6

Step-by-step explanation:

In this question, the | symbol means "given", so we can phrase the question as "Find the probability that a student watches more than one hour of TV given that they are in the 6th period class"

Next, because there is a 100% chance that the student is in the 6th period class, we can disregard the results of the 3rd period class.

Given that the student is in the 6th period class, there are 15 total students (as 9+6=15 = the sum of the yes and no answers for 6th period) and 9 students that said yes. Therefore, there is a 9/15 = 3/5 = 0.6 probability that a student watches more than 1 hour of TV given that they are in the 6th period class

Answer:

(3x+2) by (2x+1)

Step-by-step explanation:

A cardboard is a rectangle, and has two dimensions. Given a quadratic equation, you should find a way to split it in two.

The easiest way to do so is through factoring. (There are many ways to do this, take a look at the plethora of sources offered on the internet.)

When the expression 6x^2 + 7x + 2 is factored, it is (3x+2)(2x+1). Hence, these are your dimensions.

Answer:

Step-by-step explanation:

This is nice and simple. I'm going to walk through it like I do when teaching this concept to my class for the first time. This is a good problem for that.

We are given a square and we are looking for the rate at which the area is increasing when a certain set of specifics are given. That means that the main equation for this problem is the area of a square, which is:

[tex]A=s^2[/tex] where s is a side.

Since we are looking for the rate at which the area is changing, [tex]\frac{dA}{dt}[/tex], we need to take the derivative of area formula implicitly:

[tex]\frac{dA}{dt}=2s\frac{ds}{dt}[/tex] that means that if [tex]\frac{dA}{dt}[/tex] is our unknown, we need values for everything else. We are given that the initial area for the square is 49. That will help us determine what the "s" in our derivative is. We plug in 49 for A and solve:

[tex]49=s^2[/tex] so

s = 7

We are also given at the start that the sides of this square are increasing at a rate of 8cm/s. That is [tex]\frac{ds}{dt}[/tex]. Filling it all in:

[tex]\frac{dA}{dt}=2(7)(8)[/tex] and

[tex]\frac{dA}{dt}=112\frac{cm^2}{s}[/tex]

The surface area of a square of side L is given by

[tex]A = L^2[/tex]

The rate of change of the area per unit time is

[tex]\dfrac{dA}{dt} = 2L\dfrac{dL}{dt}[/tex]

We can express the length L on the right hand side in terms of the area A [tex](L = \sqrt{A})[/tex]:

[tex]\dfrac{dA}{dt} = 2\sqrt{A}\dfrac{dL}{dt}[/tex]

[tex]\:\:\:\:\:\:\:=2(\sqrt{49\:\text{cm}^2})(8\:\text{cm/s})[/tex]

[tex]\:\:\:\:\:\:\:=112\:\text{cm}^2\text{/s}[/tex]

Answer:

I think the answer is 5/6

Step-by-step explanation:

There are three even numbers and two uneven numbers less than four. Therefore, on a standard die, the.probability of Rollin a number that is even or less than for is 5/6.

Answer:

Step-by-step explanation :

Let % increase in administrative secretary be = x

Let % increase in new faculty receive be = y

Administrative secretary salary = 28,000

New faculty receive Salary = 40,000

(8)*(x/100)* (28000) + (7)*(y/100)*(40000) = 5,000

2240x +2800 y = 5,000

224x +280 y = 500

56x +70y = 125

Therefore, x and y should be chosen such that it satisfy the above equation.

The question is incomplete. The complete question is :

In a​ poll, 1100 adults in a region were asked about their online vs.​ in-store clothes shopping. One finding was that 43​% of respondents never​ clothes-shop online. Find and interpret a ​95% confidence interval for the proportion of all adults in the region who never​ clothes-shop online.

Solution :

95% confidence interval for p is :

[tex]$\hat p - Z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}} < p < \hat p + Z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}}$[/tex]

[tex]$0.43 - 1.96\times \sqrt{\frac{0.43(1-0.43)}{1100}} < p < 0.43 + 1.96\times \sqrt{\frac{0.43(1-0.43)}{1100}}$[/tex]

0.401 < p < 0.459

Therefore, 95% confidence interval is from 0.401 to 0.459

Answer:

5 ft

Step-by-step explanation:

The formula for Volume is V=lwh, or Volume = length x width x height.

The equation would be:

[tex]180=l(4)(9)[/tex]

or

[tex]180=36l[/tex]

To find the answer, divide by 36.

[tex]\frac{180}{36} =\frac{36l}{36}[/tex]

[tex]5=l[/tex]

Answer:

[tex]7x {}^{2} + 5x[/tex]

Step-by-step explanation:

[tex]9x {}^{2} + 8x - (2x {}^{2} + 3x) \\ \\ = 9x {}^{2} + 8x - 2x {}^{2} - 3x (remove \: brackets) \\\ \\ = 7x {}^{2} - 5x [/tex]

Answer:

u = 2

Step-by-step explanation:

Since this is a right triangle, we can use trig functions

cos theta = adj side / hypotenuse

cos 45 = sqrt(2) / u

u cos 45 = sqrt(2)

u = sqrt(2) / cos 45

u = sqrt(2) / 1/ sqrt(2)

u = sqrt(2) * sqrt(2)

u =2

u=2

Answer:

Solution given:

Cos 45°=base/hypotenuse

[tex]\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{u}[/tex]

doing crisscrossed multiplication

[tex]\sqrt{2}*\sqrt{2}=1*u[/tex]

u=2

Answer:

didn't get the question did u forget to put the sequence ???????

pls write the question fully so that I can help you

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Explanation:

Since f(x) is linear, this means f(x) = mx+b

Let's plug in x = 6

[tex]f(x) = mx+b\\f(6) = m*6+b\\f(6) = 6m+b[/tex]

Repeat for x = 2

[tex]f(x) = mx+b\\f(2) = m*2+b\\f(2) = 2m+b[/tex]

Now subtract the two function outputs

[tex]f(6)-f(2) = (6m+b)-(2m+b)\\f(6)-f(2) = 6m+b-2m-b\\f(6)-f(2) = 4m\\[/tex]

The b terms cancel out which is very handy.

Set this equal to 12, since f(6)-f(2) = 12, and solve for m

[tex]f(6)-f(2) = 12\\4m = 12\\m = 12/4\\m = 3\\[/tex]

So the slope of f(x) is m = 3

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Next, plug in x = 12

[tex]f(x) = mx+b\\f(12) = m*12+b\\f(12) = 12m+b[/tex]

We can then say:

[tex]f(12)-f(2) = (12m+b)-(2m+b)\\f(12)-f(2) = 12m+b-2m-b\\f(12)-f(2) = 10m\\[/tex]

Lastly, we plug in m = 3

[tex]f(12)-f(2) = 10m\\f(12)-f(2) = 10*3\\f(12)-f(2) = 30\\[/tex]

Answer:  200 miles

Work Shown:

(1.25 inches)/(250 miles) = (1 inch)/(x miles)

(1.25)/(250) = 1/x

1.25x = 250*1 ..... cross multiplication

1.25x = 250

x = 250/(1.25)

x = 200 miles

That's a question about exponentiation.

Answer:

Kenji is wrong because he does not aply the porperty correctly.

Step-by-step explanation:

A exponetiation has this form:

[tex]\boxed{a^b}[/tex]

a is the base

b is the power or exponent

To understand that situation it's important to know a property about exponentiation. When we have a multiplication with the same exponent and diferent bases, the result is the multiplication of the bases with the same exponent. Let's see this above, in mathematical language:

[tex]\boxed{a^b \cdot c^b = (a\cdot c) ^b}[/tex]

Examples:

Now, we can say why Kenji is wrong. It's easy simplify [tex]3^5 \cdot 4^5[/tex]! We know that the result is [tex](3 \cdot 4) ^5 = 12^5[/tex], but Kenji multiplied the bases and added the exponents, that's why he is wrong.

I hope I've helped. ^^

Enjoy your studies! \o/

9514 1404 393

Explanation:

There are several ways you can go at this. Here are a couple. All proofs will start with the given relations being repeated as part of the proof. Here are the next steps.

∠AED ≅ ∠BAE . . . . alternate interior angles are congruent

∠AED +∠AEB = 90° . . . . angle sum theorem

∠BAE +∠AEB = 90° . . . . substitution property of equality

∠CEF ≅ ∠BCE . . . . alternate interior angles are congruent

∠CEF +∠CEB = 90° . . . . angle sum theorem

∠BCE +∠CEB = 90° . . . . substitution property of equality

∠BAE +∠AEB = ∠BCE +∠CEB . . . . substitution property of equality

∠BAE +∠AEB = ∠BCE +∠AEB . . . . substitution property of equality

∠BAE = ∠BCE . . . . addition property of equality

∠ABE = ∠CBE = 90° . . . . BE ⊥ AC

BE ≅ BE . . . . reflexive property of congruence

ΔBEA ≅ ΔBEC . . . . ASA congruence theorem

∠BAE ≅ ∠BCE . . . . CPCTC

Answer:

The point at the bottom has to move over 2 to the left to be aligned with the point at the top however they will have a 3 space in between the 2 same for the point at the top, the top point moves over 2 to the right to be aligned with the bottom point, then they will have a 3 square space between each other.

Answer:Around 3 spaces between?

Step-by-step explanation:

Answer:

See answers below

Step-by-step explanation:

Given the following functions:

r(x) = x - 6

s(x) = 2x²

r(s(x)) = r(2x²)

Replacing x with 2x² in r(x) will give;

r(2x²) = 2x² - 6

r(s(x)) = 2x² - 6

(r-s)(x) = r(x) - s(x)

(r-s)(x) = x - 6 - 2x²

Rearrange

(r-s)(x) = - 2x²+x-6

(r+s)(x) = r(x) + s(x)

(r-s)(x) = x - 6 + 2x²

Rearrange

(r-s)(x) = 2x²+x-6

Answer:

21

Step-by-step explanation:

5/3=35/x

3 x 35=105

5 x x= 5x

105=5x

105/5=5x/5

21=x

The number of people who bought the more expensive ticket is 21.

Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Given that:-

Let the number of people be X so we can form the following expression given below:-

5/3=35/x

3 x 35=105

5 x x= 5x

105=5x

105/5=5x/5

21=x

Therefore the number of people who bought the more expensive ticket is 21.

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