Complete the square.
[tex]z^4 + z^2 - i\sqrt 3 = \left(z^2 + \dfrac12\right)^2 - \dfrac14 - i\sqrt3 = 0[/tex]
[tex]\left(z^2 + \dfrac12\right)^2 = \dfrac{1 + 4\sqrt3\,i}4[/tex]
Use de Moivre's theorem to compute the square roots of the right side.
[tex]w = \dfrac{1 + 4\sqrt3\,i}4 = \dfrac74 \exp\left(i \tan^{-1}(4\sqrt3)\right)[/tex]
[tex]\implies w^{1/2} = \pm \dfrac{\sqrt7}2 \exp\left(\dfrac i2 \tan^{-1}(4\sqrt3)\right) = \pm \dfrac{2+\sqrt3\,i}2[/tex]
Now, taking square roots on both sides, we have
[tex]z^2 + \dfrac12 = \pm w^{1/2}[/tex]
[tex]z^2 = \dfrac{1+\sqrt3\,i}2 \text{ or } z^2 = -\dfrac{3+\sqrt3\,i}2[/tex]
Use de Moivre's theorem again to take square roots on both sides.
[tex]w_1 = \dfrac{1+\sqrt3\,i}2 = \exp\left(i\dfrac\pi3\right)[/tex]
[tex]\implies z = {w_1}^{1/2} = \pm \exp\left(i\dfrac\pi6\right) = \boxed{\pm \dfrac{\sqrt3 + i}2}[/tex]
[tex]w_2 = -\dfrac{3+\sqrt3\,i}2 = \sqrt3 \, \exp\left(-i \dfrac{5\pi}6\right)[/tex]
[tex]\implies z = {w_2}^{1/2} = \boxed{\pm \sqrt[4]{3} \, \exp\left(-i\dfrac{5\pi}{12}\right)}[/tex]
solve the system of equations using substitution
The solution to the given simultaneous equation; x + 2y = -4, 2x + 5y = 1 using substitution method is x = -22 and y = 9
Simultaneous equationx + 2y = -4
2x + 5y = 1
Using Substitution method
From equation (1)
x = -4 - 2y
Substitute x = -4 - 2y into equation (2)
2x + 5y = 1
2(-4 - 2y) + 5y = 1
-8 - 4y + 5y = 1
- 4y + 5y = 1 + 8
y = 9
Substitute y = 9 into
x + 2y = -4
x + 2(9) = -4
x + 18 = -4
x = -4 - 18
x = - 22
Therefore, the solution to the given simultaneous equation; x + 2y = -4, 2x + 5y = 1 using substitution method is x = -22 and y = 9
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12. The base of a triangle with an area of 36 squared inches is 4.2 inches. What is the area of a similar
triangle whose base measures 5.6 inches?
The 4.2 inches base length and 36 in.² area of the given triangle and the 5.6 inches base length of the similar triangle gives the area of the similar triangle as 64 square inches
Which method can be used to find the area of the similar triangle given the dimensions?Area of a triangle = (Base length × Height)/2
Area of the given triangle = 36 in.²
Base length of the given triangle = 4.2 inches
Base length of the similar triangle = 5.6 inches
Therefore;
Area of the given triangle = (Base length × Height)/2
Which gives;
36 = (4.2 × h)/2
Where;
h = Height of the given triangle
36 × 2 = 4.2 × h
[tex]h = \mathbf{\frac{36 \times 2}{4.2}} = 17 \frac{1}{7} [/tex]
Height of the given triangle, h = 17+ 1/7
The ratio of corresponding sides of similar triangles are the same, which gives;
[tex] \frac{5.6}{4.2} = \frac{h'}{17 \frac{1}{7}} [/tex]
Where;
h' = The height of the similar triangle
Which gives;
[tex] h' = \frac{5.6}{4.2} \times 17 \frac{1}{7} = 22 \frac{6}{7} [/tex]
The area, A', of the similar triangle is therefore;
[tex] A' = \frac{1}{2} \times 5.6 \times 22 \frac{6}{7} = 64 [/tex]
The area of the similar triangle A' = 64 in.²The area can also be obtained using the scale factor of area as follows;
(4.2/5.6)² = 36/A'Which gives;
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If 55 people enrolled in a psychology course but only 25 completed it, what percent of the students completed the course? (Round to the nearest tenth of a percent.)
Answer:
45%
Step-by-step explanation:
Simply, 25/55 which gives you 0.454545.
Convert that to a percentage and you get 45%
In the figure below, the segment is parallel to one side of the triangle. Find the value of x.
x+7
18²/3
0 24
025²/
16
22
Step-by-step explanation:
answer=18^2/3
according to the congruent triangles are present there.
so divide corresponding sides
50% prefferd fruit cake 1/5 preferred spong and the remainder preferred cheesecake
What percent like cheesecake?
The percentage of people who preferred cheesecake if 50% prefferd fruit cake 1/5 preferred sponge is 40%.
PercentageTotal percentage = 100%Fruit cake = 50%Percentage remaining = 100% - 50%
= 50%
Sponge cake = 1/5
Percentage of sponge cake = 1/5 × 50%
= 1/5 × 0.5
= 10%
Percent that likes cheesecake = Total - (fruits cake + sponge cake)
= 100% - (50% + 10%)
= 100% - (60%)
= 40%
Therefore, the percentage of people who preferred cheesecake if 50% prefferd fruit cake 1/5 preferred sponge is 40%.
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Four seniors and six juniors are competing for four places on a quiz bowl team.
What is the approximate probability that all four seniors will be chosen at random?
what’s the % ?
Answer:
.00476
Step-by-step explanation:
Use the coordinate grid to determine the coordinates of point A:
Coordinate grid shown from negative 2 to positive 2 on x axis and negative 2 to positive 2 on y axis. There are increments of 1 over 4 for each grid line on each of the two axes. Only the whole numbers are labeled on either side of the axis. A point A is shown at the intersection of 5 grid lines to the left of the y-axis and 3 grid lines above the x-axis.
What are the coordinates of point A?
( fraction negative 1 and 1 over 4, fraction 3 over 4)
( fraction 1 and 1 over 4, fraction negative 3 over 4)
( fraction negative 3 over 4, fraction 1 and 1 over 4)
( fraction 3 over 4, fraction negative 1 and 1 over 4)
Based on the given coordinate grid, and the increment for each grid line, the coordinates of point A is ( fraction negative 1 and 1 over 4, fraction 3 over 4) or (-1¹/₄, ³/₄).
What are point A's coordinates?
The x-value of point A is said to be 5 grid lines to the left of y-axis. This means that the x value will be negative.
The increments are 1/4 so the x-value is:
= -1/4 x 5
= -1¹/₄
The y-value is given by the 3 grid lines above the x-axis which means that this value is positive:
= 1/4 x 3
= 3 /4
The coordinates of A are:
(-1¹/₄, ³/₄)
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Knowing the first day Anna and Tamara stop for sightseeing and lost 2 hours of travel time and the second day, they gain 1 hour because they did not stop for lunch, write an algebraic expression that represents how many hours Anna and Tamara traveled the first two days. Let x be the number of hours driven.
The algebraic expression that represents the number of hours Anna and Tamara traveled the first two days is (2x - 1) hours.
How to write Algebraic Word Problems?
We are told that;
On the first day, Anna and Tamara stop for sightseeing and lost 2 hours of travel time.
On the second day, they did not stop for lunch and gained 1 hour.
Now, we are told that x is the number of hours driven per day. Thus, for two days, number of hours driven is expressed as 2x.
Now, since they lost 2 hours on the first day, then total hours travelled is;
2x - 2 hours
On the second day, they gained 1 hour. Thus;
Final total number of hours traveled = 2x - 2 + 1
Final total number of hours traveled = (2x - 1) hours
Thus, we can conclude that the algebraic expression that represents how many hours Anna and Tamara traveled the first two days is (2x - 1) hours.
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30% of 70 is what percent of 2000?
Answer:
1.05
Step-by-step explanation:
Because 30% of 70 is [tex]30\% \times 70 = \frac{30}{100} \times 70 = 0.3 \times 70 = 21[/tex], so the question becomes "21 is what percent of 2000?"
To answer this, we can solve for [tex]x[/tex] in the equation [tex]21 = x\% \times 2000[/tex], which is:
[tex]21 = x\% \times 2000\\21 = \frac{x}{100} \times 2000\\21 = 20x\\x=\frac{21}{20}=1.05[/tex]
So the answer is 1.05 percent.
I need help with the problem
Answer:
you better give me brainliest
Step-by-step explanation:
The answer is the first two
At the beginning of this year, you went out and purchased $300 worth of gold coins. Every 3 months for the next 8 years, you purchase an additional $300 worth of coins. At the end of the first 3 years, the price of gold starts to come back down, and when you decide to cash out, your coins are only worth what you paid for them. What is the value of your account now?
Answer:
9900
Step-by-step explanation:
You start with 300, and add 300 32 times (once for every 3 months in 8 years).
[tex]300+9600=9900[/tex]
the confidence interval for the population variance is
(730.72, 4569.53) is the confidence interval for the population variance σ² given that c = 0.98, s = 39 and n = 15. This can be obtained by using formula for confidence interval and using statistical table for values.
Find the confidence interval for population variance:The formula for finding the confidence interval for population variance is:
[tex]\frac{(n-1)s^{2} }{\chi^{2} _{\frac{\alpha }{2} } } < \sigma^{2} < \frac{(n-1)s^{2} }{\chi^{2} _{1-\frac{\alpha }{2} } }[/tex]
where n is the size of the sample, (n - 1) is the degrees of freedom, s is the standard deviation, α is the significance level.
Here it is given that,
confidence level = 1 - α = 0.98 significance level = α = 0.02standard deviation = 39 sample size n = 15degrees of freedom (n - 1) = 15 - 1 = 14α/2 = 0.02/2 = 0.01
1 - α/2 = 1 - 0.01 = 0.99
For (n - 1) degrees of freedom,
[tex]\chi^{2}_{\frac{\alpha}{2} }[/tex] = 29.141 , and [tex]\chi^{2}_{1-\frac{\alpha}{2} }[/tex] = 4.660 (Using statistical table)
Using the formula we get,
[tex]\frac{(n-1)s^{2} }{\chi^{2} _{\frac{\alpha }{2} } } < \sigma^{2} < \frac{(n-1)s^{2} }{\chi^{2} _{1-\frac{\alpha }{2} } }[/tex]
[tex]\frac{(14)39^{2} }{29.141} < \sigma^{2} < \frac{(14)39^{2} }{4.660}[/tex]
[tex]730.72 < \sigma^{2} < 4569.53[/tex]
The confidence interval for the population variance is (730.72, 4569.53)
Hence (730.72, 4569.53) is the confidence interval for the population variance σ² given that c = 0.98, s = 39 and n = 15.
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If a translation maps point A(-3,1) to point A′(5,5), the translation is:
A. (x+2,y+4)
B. (x+8,y+6)
C. (x+2,y+6)
D. (x+8,y+4)
Answer:
D is the correct answer -3+8=5 ,1+4=5
A bookstore carries 48 magazines. 75% of the magazines are women's magazines. How many women's magazines does the bookstore carry?
Answer:
36
Step-by-step explanation:
75/100
75/100=0.75
0.75*48 =36
K
An employment agency specializing in temporary construction help pays heavy equipment operators $124 per day and general laborers $82 per day If thirty-seven people were hired
and the payroll was $3916, how many heavy equipment operators were employed? How many laborers?
The number of heavy equipment operators hired was?
The number of general laborers hired was?
Step-by-step explanation:
x = number of heavy equipment operators
y = number of general laborers
x + y = 37 (37 people were hired)
out if this we get e.g.
x = 37 - y
124x + 82y = 3916 (daily rate for each role, in total 3916)
now we use the first equation in the second :
124(37 - y) + 82y = 3916
4588 - 124y + 82y = 3916
672 - 42y = 0
672 = 42y
y = 16
x = 37 - y = 37 - 16 = 21
21 heavy equipment operators were hired.
16 general laborers were hired.
A car is traveling at a speed of 60 miles per hour. What's the dependent variable in this situation? Question 5 options: A) The speed at which the car travels B) The distance the car has traveled C) The age of the car D) The number of hours the car has traveled
The dependent variable in this situation is; the speed at which the car travels.
Distance and speedSpeed of the car = 60 miles per hourSpeed = Distance / Time
The dependent variable in this situation is; the speed at which the car travels.
The independent variable; The distance the car has traveled.
What is dependent variable?This is the variable whose value depends on one or more variables in the equation.
Independent variable
This is the variable whose value is not dependent on any other in the equation.
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write the equations after translating the graph of y=|x+2| one unit to the right
write the equations after translating the graph of y=|x+2| one unit to the left
What is the length of S?
Answer:
c
Step-by-step explanation:
By the Pythagorean theorem,
[tex]28^2 + 15^2 = s^2 \\ \\ s^2 = 1009 \\ \\ s = \sqrt{1009}[/tex]
Which is a factor of 144 - 49x²?
O 12-7x²
O 72-7x²
O 12-7x
O 72 + 7x
Answer: C: 12-7x
Step-by-step explanation:
This expression is a good example of the difference of two squares, as 144 is the square of 12, while 49x² is the square of 7x.
Any difference of two squares [tex]a^2-b^2[/tex] can be factored into the sum of their square roots times the difference of their square roots [tex](a+b)(a-b)[/tex].
Let's factor our expression.
[tex]144-49x^2=(12+7x)(12-7x)[/tex]
Out of all the options given, only [tex]12-7x[/tex] is a factor. Hence, option C is correct.
Anna does sit-ups to get ready for her first triathlon. When she starts, she does a sit-up every 2 seconds. But, as she gets tired, each sit-up takes longer to do. Is the number of sit-ups Anna does proportional to the time she spends doing them?
Answer: No
Step-by-step explanation: proportional means corresponding in size or amount to something. However, the number of her sit-ups isn't consistent, so it isn't proportional to the time she spends doing them.
Please answer this question
Answer:
1 and 4
Step-by-step explanation:
x-1 = 0
x = 1
x^2 - 5x + 4
(x-4)(x-1) = 0
x can't be 1 or 4
Answer: 1 and 4
Step-by-step explanation:
1. To find excluded values, we need to make the denominator equal to zero.
x² - 5x + 4 = 0
2. Then we need to find the x the product of them was 4, but they add up to -5, which are -4 and -1. We get:
(x - 4)(x - 1) = 0
x - 4 = 0 x - 1 = 0
x = 4 x = 1
These two numbers will make the denominator equal to 0, the denominator cannot be equal to 0, so they are excluded numbers.
In the following activity, match each pair of equivalent expressions(will give 50 points)
Identify the theorem or postulate that is related to the measures of the angles in the pair, and find the unknown angle measure.
m∠1=(50x−50)∘, m∠2=(20x+70)∘
Alt. Ext. ∠s Thm.
m∠1 = 130°, m∠2 = 130°
Same-Side Int. ∠s Thm.
m∠1 = 130°, m∠2 = 130°
Alt. Int. ∠s Thm.
m∠1 = 150°, m∠2 = 150°
Corr. ∠s Post.
m∠1 = 150°, m∠2 = 150°
The theorem used and the angle measures are:
D. corresponding angles postulate
m∠1 = 150°, m∠2 = 150°
What is the Corresponding Angles Postulate?According to the corresponding angles postulate, if two angles lie on same relative corner along a transversal that cuts two parallel lines (corresponding angles), then their measures are equal.
Thus, ∠1 and ∠2 are corresponding angles, therefore:
m∠1 = m∠2 [corresponding angles postulate]
Given the following:
m∠1 = (50x − 50)∘,
m∠2 = (20x + 70)∘
Therefore, we would have:
50x − 50 = 20x + 70
Subtract 20x from both sides
50x − 50 - 20x = 20x + 70 - 20x
30x − 50 = 70
Add 50 to both sides of the equation
30x − 50 + 50 = 70 + 50
30x = 120
Divide both sides by 30
30x/30 = 120/30
x = 4
Plug in the value of x
m∠1 = (50x − 50) = 50(4) − 50 = 150°
m∠2 = (20x + 70)∘ = 20(4) + 70 = 150°
Thus, the answer is: D. corresponding angles postulate
m∠1 = 150°, m∠2 = 150°
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Which of the following describes the transformations of g (x) = negative (2) Superscript x + 4 Baseline minus 2 from the parent function f (x) = 2 Superscript x?
Using translation concepts, the transformations are described as follows:
Reflection over the x-axis.Shifted up 4 units.What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
In this problem, the parent function is:
[tex]f(x) = 2^x[/tex]
The transformed function is:
[tex]g(x) = -2^x + 4[/tex]
We have that the transformations in this problem can be described as follows:
f(x) was multiplied by -1, hence it was reflected over the x-axis.4 was added to the function, hence it was shifted up 4 units.More can be learned about translation concepts at https://brainly.com/question/28098112
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Given angle YMC = 170 degrees, find the measure of angle AMC.
Answer:
i think 10 because in triangle there is 180 and 180-170 = 10
Answer:
Step-by-step explanation:
hjhj
Find the remainder when f(x)=−2x3+x2−4x+1 is divided by x+3.
The remainder when f(x)=−2x3+x2−4x+1 is divided by x+3. is 76
How to determine the remainder?The function is given as:
f(x)=−2x3+x2−4x+1 is divided by x+3.
Set the divisor to 0
x + 3 = 0
Solve for x
x = -3
Calculate f(-3)
f(-3)=−2(-3)^3+(-3)^2 − 4(-3) + 1
Evaluate the expression
f(-3)= 76
Hence, the remainder when f(x)=−2x3+x2−4x+1 is divided by x+3. is 76
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PLS HELP
Which functions have a range of {y e R-00 < y < 00}?
O f(x)
2x+3
Of(x) = x - 8
O f(x) = x² + 7x - 9
O f(x) = -4x + 11
O f(x) = -(x + 1)² - 4
=
Answer:
[tex]\displaystyle{1.) \ \ f(x) = \dfrac{2}{3}x-8}\\\\\displaystyle{2.) \ \ f(x) = -4x+11}[/tex]
Step-by-step explanation:
- Range is a set of all y-values in a set of coordinate points.
A linear equation or function always have range equal to set of real number because you can substitute f(x) as any numbers and you'll still be able to solve for x-variable.
An quadratic equation or function may have range equal to set of positive real number or negative real number depending whether if the coefficient of x² is in positive or negative but consider this:
Suppose we have f(x) = x², f(x) cannot be negative number because that'd make the equation not real. Therefore, a quadratic function does not have [tex]\displaystyle{\mathbb{R}}[/tex] range.
For an exponential function, it's same as quadratic equation. It depends whether if a base is in negative or positive. You can consider like this:
Suppose we have [tex]\displaystyle{f(x)=2^x}[/tex], f(x) cannot be negative number because there are no x-values that make the equation true. Therefore, an exponential function does not have [tex]\displaystyle{\mathbb{R}}[/tex] range as well.
Between which two integers is the value of √11?
9 and 10
5 and 6
7 and 8
O3 and 4
1 and 2
PLS HELP ME ASAP I NEED THE ANSWER NOW BEFORE 5 mins PLS Help ME I WILL MARK BRAINLIST IF YOUR ANSWER IS CORRECT AND WAS ANSWERED 4 MIN AGO HELP ME PLSSS
The table below shows that 1 day equals 24 hours. Notice that 12 hours is one-half of that. Use the table to help you determine how many days are equal to 42 hours.
1 day
24 hours
12 hours 12 hours
1 and 1 over 4. days (wrong)
1 and 1 over 2. days
1 and 3 over 4. days
2 days
Answer:
42 - 24 = 18
18/24 = 3/4
the answer would be 1 and 3/4 days
Answer:
1 and 3 over 4 days
Step-by-step explanation:
Since 24 hours is 1 day all we have to do is divide 42 into 24 and see what the resulting fraction is
42/24 = 7/4 (reducing to lowest fraction)
7/4 = 1 3/4 days (7 ÷ 4 gives quotient 1 and remainder is 3)
Answer:
1 and 3 over 4 days
ASAP help me with this question <3 number 12
Answer:
False
Step-by-step explanation:
Consider an isoscles trapezoid with angles 75° and 105°.