By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.
To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).
The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.
Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.
However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:
f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)
To know more about rational zeros theorem:
https://brainly.com/question/29004642
#SPJ4
4) 6 out of the 80 dogs in a shelter were adopted yesterday. Express adopted rate as a percent.
Answer:
Step-by-step explanation:
To find the adoption rate as a percentage, we need to divide the number of dogs adopted by the total number of dogs in the shelter, then multiply by 100.
adoption rate = (dogs adopted / total dogs) * 100%
adoption rate = (6 / 80) * 100%
adoption rate = 0.075 * 100%
adoption rate = 7.5%
Therefore, the adoption rate as a percent is 7.5%.
In which number is the figit 8 ten times it is in the number 18?
Answer:
Let's call the number we are looking for "x". We are told that the digit 8 appears ten times as often in "x" as it does in 18.
The digit 8 appears once in the number 18, so it must appear 10 times in "x".
Let's count the number of 8's in "x" in terms of the number of digits of "x".
If "x" has 1 digit, then it cannot have 10 8's, so we can rule out this case.
If "x" has 2 digits, then the maximum number of 8's it can have is 9 (e.g., 88). This is still not enough, so we can rule out this case as well.
If "x" has 3 digits, then the maximum number of 8's it can have is 27 (e.g., 888), which is enough.
Therefore, the number we are looking for is a 3-digit number that contains 10 8's. We can write such a number as:
x = 888x8x8x8
where "x" can be any digit other than 8.
Note that there are other 3-digit numbers that contain 10 8's, but this is one possible solution.
(please mark my answer as brainliest)
xercise 1.3.4 in each case, either express y as a linear combination of a1, a2, and a3, or show that it is not such a linear combination. here:
Here, y can be expressed as a linear combination of a1, a2, and a3 is y = (-1/8) a1 + (9/4) a2 - (5/2) a3.
We can express y as a linear combination of a1, a2, and a3 if and only if y is a linear combination of the column vectors of the matrix A whose columns are a1, a2, and a3. We can write this as:
y = c1 a1 + c2 a2 + c3 a3
where c1, c2, and c3 are constants to be determined. We can solve for these constants by writing the system of equations in matrix form:
A [c1; c2; c3] = y
where [c1; c2; c3] is a column vector of the constants c1, c2, and c3. We can solve for [c1; c2; c3] by multiplying both sides by the inverse of A (assuming it exists):
[c1; c2; c3] = A^(-1) y
If A^(-1) exists, then y can be expressed as a linear combination of a1, a2, and a3. Otherwise, y cannot be expressed as a linear combination of a1, a2, and a3.
For y = [1 2 4 0], we have:
A = [-1 3 0 1; 3 1 2 0; 1 1 1 1]
We can compute the inverse of A using row reduction:
[A | I] = [-1 3 0 1 | 1 0 0;
3 1 2 0 | 0 1 0;
1 1 1 1 | 0 0 1]
[R2 - 3R1, R3 - R1] = [-1 3 0 1 | 1 0 0;
0 -8 2 -3 | -3 1 0;
0 -2 1 0 | -1 0 1]
[R2 / (-8), R3 + 2R2] = [1/8 -3/8 0 3/8 | 3/8 -1/8 0;
0 1 0 -1/4 | 3/4 -1/4 0;
0 0 1 -1/2 | 1/2 -1/2 1]
Therefore, A^(-1) = [1/8 -3/8 0 3/8;
0 1 0 -1/4;
0 0 1 -1/2;
0 0 0 0]
We can now compute [c1; c2; c3]:
[c1; c2; c3] = A^(-1) y = [1/8 -3/8 0 3/8;
0 1 0 -1/4;
0 0 1 -1/2;
0 0 0 0] [1; 2; 4; 0] = [-1/8; 9/4; -5/2; 0]
Therefore, y as a linear combination of a1, a2, and a3:
y = (-1/8) a1 + (9/4) a2 - (5/2) a3
To know more about linear combination here
https://brainly.com/question/30326131
#SPJ4
_____The given question is incomplete, the complete question is given below:
Exercise 1.3.4 in each case, either express y as a linear combination of a1 = [-1 3 0 1], a2 = [3 1 2 0], and a3= [1 1 1 1], or show that it is not such a linear combination. here: y = [1 2 4 0]
need some help on some questions
For the triangle ABC, the given trigonometric ratios are -
a. sin A = 8/17
b. cos A = 15/17
c. tan A = 8/15
d. tan B = 8/15
What is trigonometric ratio?
Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.
For a right-angled triangle ABC, the hypotenuse AB is given as 17.
The base CB is given as 15 and the perpendicular AC is given as 8.
The angle C is given to be 90°.
Using the given values of the sides of the right triangle ABC, we can calculate the trigonometric ratios as follows -
a. sin A = opposite/hypotenuse = AC/AB = 8/17 (reduced fraction)
b. cos A = adjacent/hypotenuse = CB/AB = 15/17 (reduced fraction)
c. tan A = opposite/adjacent = AC/CB = 8/15 (reduced fraction)
d. tan B = opposite/adjacent = AC/CB = 8/15 (reduced fraction)
Note that since angle C is 90°, angles A and B are acute angles, so their tangent ratios are equal to each other.
Therefore, the ratios expressed as reduced fractions are -
a. sin A = 8/17
b. cos A = 15/17
c. tan A = 8/15
d. tan B = 8/15
To learn more about trigonometric ratio from the given link
https://brainly.com/question/13276558
#SPJ1
FI;LL IN THE BLANK. An online retailer has determined that the average time for credit card transactions to be electronically approved K 1.6 seconds. (Round your answers to three decimal places.) (a) Use on exponential density function to find the probability that a customer warts less than a second for credit card approval. (b) Find the probability that a customer waits more than 3 seconds. ____ (c) What Is the minimum approval time for the slowest 5% of transactions? ____sec
(a) By using the exponential density function, the probability that a customer warts less than a second for credit card approval is 0.334
(b) The probability that a customer waits more than 3 seconds is 0.154
(c) The minimum approval time for the slowest 5% of transactions is 4.013 seconds
(a) To find the probability that a customer waits less than a second for credit card approval, we need to use the exponential density function:
f(x) = [tex]\lambda e^{-\lambda x}[/tex]
Where λ is the rate parameter, which in this case is the reciprocal of the mean approval time. So, λ = 1/1.6 = 0.625.
The probability that a customer waits less than a second can be calculated by integrating the density function from 0 to 1:
P(X < 1) = ∫[tex]0^1 \lambda e^{-\lambda x}[/tex] dx
P(X < 1) = [tex][-e^{-\lambda x}]0^1[/tex]
P(X < 1) = [tex]-e^{(-0.625)}[/tex] + 1
P(X < 1) = 0.334
Therefore, the probability that a customer waits less than a second for credit card approval is 0.334.
(b) To find the probability that a customer waits more than 3 seconds for credit card approval, we can use the same exponential density function and integrate from 3 to infinity:
P(X > 3) = ∫[tex]3^{\infty} \lambda e^{-\lambda x}[/tex] dx
P(X > 3) = [[tex]-e^{-\lambda x}[/tex])][tex]3^{\infty}[/tex]
P(X > 3) = [tex]e^{-1.875}[/tex]
P(X > 3) = 0.154
Therefore, the probability that a customer waits more than 3 seconds for credit card approval is 0.154.
(c) We can use the exponential distribution's inverse function to find this value:
P(X > x) = 0.05
[tex]e^{-\lambda x}[/tex] = 0.05
-xλ = ln(0.05)
x = ln(0.05)/(-λ)
x = ln(0.05)/(-0.625)
x = 4.013 seconds
Therefore, the minimum approval time for the slowest 5% of transactions is 4.013 seconds.
To know more about probability here
https://brainly.com/question/11234923
#SPJ4
Question 19 (2 points)
According to research conducted by the Department of Education, 80% of college
students took a mathematics course as part of their general education requirements.
If ten college students are selected at random, what is the probability at least one of
the ten has not taken a mathematics course?
0.0800
0.1073
0.7927
0.8000
Answer:
The probability that a single college student has not taken a mathematics course is 1 - 0.8 = 0.2.
The probability that all ten selected college students have taken a mathematics course is (0.8)^10 = 0.1074 (rounded to four decimal places).
Therefore, the probability that at least one of the ten selected college students has not taken a mathematics course is:
1 - 0.1074 = 0.8926 (rounded to four decimal places).
So the answer is 0.8926, which is closest to option C (0.7927).
(please mark my answer as brainliest)
I will mark you brainiest!
Given the coordinates shown and given that SU = 10, what are the coordinates of U if STUV is a kite?
A) (10, 18)
B) (0, 28)
C) (18, 28)
The calculated coordinates of U if STUV is a kite is (10, 18)
Calculating the coordinates of U if STUV is a kite?From the question, we have the following parameters that can be used in our computation:
The figute of a kite
Also, we have
S = (0, 18)
And the distance SU to be
SU = 10
If the quadrilateral STUV is a kite, then the coordinates S and U are on the same horizontal level (according to the figure)
So, we have
U = (0 + 10, 18)
Evaluate
U = (10, 18)
Hence, the coordinates of U if STUV is a kite is (10, 18)
Read more about distance at
https://brainly.com/question/31617619
#SPJ1
How do you solve this equation?
Solved equation x=80 and z=2, y=40
What is Variables?An element, feature, οr factοr that is liable tο vary οr change
If y varies directly as x and inversely as the square οf z, we can write the fοllοwing prοpοrtiοnality:
y ∝ x/z²
where ∝ denοtes prοpοrtiοnality cοnstant.
Tο find the value οf ∝, we can use the given values οf y, x, and z:
y = ∝ x/z²
28 = ∝ (63)/(3)²
∝ = 28 * (3)² / (63)
∝ = 4/3
Nοw we can use this value οf ∝ tο find y when x=80 and z=2:
y = ∝ x/z²
y = (4/3) * (80)/(2)²
y = 40
Therefοre, when x=80 and z=2, y=40.
To learn more about Variables, click on the link below:
https://brainly.com/question/29696241
#SPJ1
Mason earns $8.10 per hour and worked 40 hours. Noah earns $10.80 per hour. How many hours would Noah need to work to equal Mason’s earnings over 40 hours?
Answer:
Noah would need to work 30 hours to equal Mason's earning for 40 hours
Step-by-step explanation:
Mason;
8.10 x 40 = 324
324 ÷ 10.80 = 30 hours.
Helping in the name of Jesus.
Answer:
30 hours
Step-by-step explanation:
40 times 8.10 is 324 and 324 divided by 10.8 is 30 hours
i need help i forgot how to do this
Step-by-step explanation:
For RIGHT triangles : sin ( angle) = opposite LEG / Hypotenuse
so sin B = 32 / 68 = 8 / 17
HELP ILL GIVE YOU ONE HUNDRED POINTS AND BE MARKED BRAINLIEST IF YOU HELP MME ITS DUE TODAY!!!!
Answer: Somewhere around 40%, write 0.4.
Step-by-step explanation:
I took all of the outcomes from the games got the. probabilities of winners and got 40%.
the area of a rectangle is 33.12cm². Given that one side of length is 4.6cm. Find the length of the other side
Answer:
7.2 cm----------------------------
Area of rectangle formula:
A = lwGiven:
A = 33.12 cm²,w = 4.6 cm.Find the missing side length:
l = A/wl = 33.12/4.6l = 7.2 cmAnswer:
7.2cm
Step-by-step explanation:
We are here given that, the area of a rectangle is 33.12cm² and one of the side is 4.6cm .
We are interested in finding the length of the other side,
As we know that the area of rectangle is calculated using the formula,
Area = l * b
where l is the length and b is the breadth.
Now substitute the respective values,
33.12cm² = l * 4.6cm
l = 33.12cm²/4.6cm
l = 7.2 cm
Hence the value of second side is 7.2cm
What is the meaning of "complex numbers with absolute value 1 "?
Answer:
A complex number is said to have an absolute value of 1 if the magnitude of its real and imaginary parts together is equal to 1. Mathematically, |z| = (a²+b²)^½ = 1, where z is the complex number with real part ‘a’ and imaginary part ‘b’.I will mark you brainiest!
How many polygon types (based on the number of sides in each) are shown in the diagram below?
A) 4
B) 3
C) 2
Answer:
A) 4
Step-by-step explanation:
There are 2 lines dividing the polygon.
The first two on the top are triangles.
The two on the bottom are irregular polygons.
Added up, that's 4 polygons according to the line distribution.
please help!! its for homework that is due very soon!!
Answer:
A>C>B
Step-by-step explanation:
The angle facing or opposite the longest side is the largest angle while.............
22 and 28 are two numbers Express the smallest number as a percentage of the sum of the two numbers
Answer: 44%
Step-by-step explanation:
22 is the smallest of the two numbers, and you want that number as a percentage of the sum of the two numbers. So basically it is asking you to put 22/(22+28) as a percentage. The step by step is as follows:
1. Simplify denominator
22+28 = 50
2. Rewrite the fraction so that the numerator is out of 100
22/50 = 44/100
3. Convert this new fraction to percentage
44/100= 44%
Hope this helps!!
NEED HELPPPP PLSSSS
DUE TOMORROW!!
8.The method that resulted in Garcia winning might not be fair to the other nominees because it does not account for their individual achievements or skill sets.
What is nominees ?Nominees are individuals or organizations that have been selected or chosen from a given group to represent them in a certain process or event. Nominees are usually chosen for their proficiency, expertise, or reputation in a certain field or area. For example, in some elections, political parties nominate individuals to represent them in the election and these nominees are then voted upon by the public. Similarly, some organizations nominate a particular individual or group of individuals to represent them in awards or recognition ceremonies for their excellence and achievements.
It simply rewards the nominee with the most votes, regardless of their ability or accomplishments. This could lead to someone who is less qualified or experienced receiving the award, while the more deserving candidates are overlooked.
To learn more about nominees
https://brainly.com/question/17268004
#SPJ1
let be the space spanned by the two functions and . find the matrix of the linear transformation from into itself with respect to the basis .
When space is spanned by the two functions of linear transformation from into itself with respect to the basis we need to apply T to each basis vector vi to get the column vectors T(vi) = [T(vi)]B.
where [T(vi)]B is the coordinate vector of T(vi) with respect to the basis B. Arrange the column vectors [T(v1)]B, [T(v2)]B, ..., [T(vn)]B into a matrix. This matrix is the matrix of T with respect to the basis B.
In this case, you have two functions that span a vector space, so you need to specify the basis B. Once you have chosen the basis, you can apply the above steps to find the matrix of the linear transformation.
To learn about linear transformation visit:
https://brainly.com/question/30585642
#SPJ4
whats the radius of the button
Answer:
2.5
Step-by-step explanation:
diameter is 5 and the radius is half of the diameter so it is 2.5
hope this helps :)
I will mark you brainiest!
CPCTC can be used in proofs after triangles have been shown to be congruent.
A) True
B) False
Answer:
True.
Step-by-step explanation:
CPCTC is commonly used at or near the end of a proof, which asks the student to show that two angles or two sides are congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be congruent.
PLEASE HELP YOU WILL BE BRAINIEST!!! Use the chart!!!!
The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 385 grams and a standard deviation of 8 grams find the weight that corresponds to each event(use excel or appendix c to calculate the z value. Round your final answers to 2 decimal places)
URGENT
the weight that corresponds to each of the events are:
a) The weight is less than 380 grams: [tex]$P(X < 380)=0.266$[/tex].
b) The weight is between 375 and 395 grams: [tex]$P(375 < X < 395)=0.7887$[/tex].
c) The weight is greater than 400 grams: [tex]$P(X > 400)=0.0304$[/tex].
How to deal with Normal distribution?Let X be the weight of a small Starbucks coffee. We are given that X is normally distributed with mean [tex]$\mu=385$[/tex] grams and standard deviation [tex]$\sigma=8$[/tex].
We want to find the weight that corresponds to each of the following events:
a) The weight is less than 380 grams.
b) The weight is between 375 and 395 grams.
c) The weight is greater than 400 grams.
To solve these problems, we first standardize the distribution by finding the corresponding z-scores using the formula:
[tex]$z=\frac{X-\mu}{\sigma}$$[/tex]
a) The weight is less than 380 grams.
We want to find P(X<380). We can find the z-score for X=380 as follows:
[tex]$z=\frac{380-385}{8}=-0.625$$[/tex]
Using a standard normal table or calculator, we find that the probability P(Z<-0.625)=0.266. Therefore,
[tex]$P(X < 380)=P\left(Z < -\frac{0.625}{1}\right)=0.266$$[/tex]
b) The weight is between 375 and 395 grams.
We want to find [tex]$P(375 < X < 395)$[/tex]. We can find the z-scores for X=375 and X=395 as follows:
[tex]$z_1=\frac{375-385}{8}=-1.25,\quad z_2=\frac{395-385}{8}=1.25$$[/tex]
Using a standard normal table or calculator, we find that the probability P(-1.25<Z<1.25)=0.7887. Therefore,
[tex]$P(375 < X < 395)=P\left(-1.25 < Z < 1.25\right)=0.7887$$[/tex]
c) The weight is greater than 400 grams.
We want to find P(X>400). We can find the z-score for X=400 as follows:
[tex]$z=\frac{400-385}{8}=1.875$$[/tex]
Using a standard normal table or calculator, we find that the probability P(Z>1.875)=0.0304. Therefore,
[tex]$P(X > 400)=P\left(Z > \frac{1.875}{1}\right)=0.0304$$[/tex]
Therefore, the weight that corresponds to each of the events are:
a) The weight is less than 380 grams: [tex]$P(X < 380)=0.266$[/tex].
b) The weight is between 375 and 395 grams: [tex]$P(375 < X < 395)=0.7887$[/tex].
c) The weight is greater than 400 grams: [tex]$P(X > 400)=0.0304$[/tex].
To know more about Normal distribution visit:
brainly.com/question/29509087
#SPJ1
explain why a set in r5 ust be linearly independent when is linearly indedependent and v4 is not in span
A set in R⁵ must be linearly independent because of the dimensionality of the space.
A Set of vectors in R⁵, the five-dimensional Euclidean space, must be linearly independent because of the dimensionality of the space.
The maximum number of linearly independent vectors in any set in R⁵ is 5 since any set with more than 5 vectors would necessarily contain a linearly dependent subset.
This is because any vector in R⁵ can be expressed as a linear combination of at most 5 linearly independent vectors, as the dimension of R⁵ is 5.
Therefore, any set with more than 5 vectors would have at least one vector that could be written as a linear combination of the other vectors in the set, making the set linearly dependent.
Learn more about Linearly Independent Sets here
https://brainly.com/question/29646890
#SPJ4
The given question is incomplete, the complete question is
Explain why a set in R⁵ must be linearly independent.
Tristan is going to invest $73,000 and leave it in an account for 18 years. Assuming
the interest is compounded continuously, what interest rate, to the nearest tenth of a
percent, would be required in order for Tristan to end up with $104,000?
Answer:
2%
Step-by-step explanation:
Given,P = $73000
A = $104000
T = 18 years, Compounded annually.
To find: r%
Soln: By formula, A = 73000*(1+r/100)^18
=> (104/73)^1/18 = (100+r)/100
=> 1.0198 = 100+r/100
=> 101.98 -100 = r
=> 1.98 = r
To the nearest percent, 2 = r
Hence, Rate of interest = 2%
d. Two judges in a beauty contest rank the ten competitors in the following order.
Do d. Two judges in a beauty contest rank the ten competitors in the following order.
Do the two judges appear to agree in their standard? the two judges appear to agree in their standard?
The correlation coefficient is close to zero, we can conclude that the two judges do not appear to agree in their standards.
What is correlation and causation in statistics?Nevertheless, a correlation between two variables does not always imply that a change in one variable is the reason for a change in the values of the other.
There is a causal link between the two occurrences, which means that causation shows that one event is the outcome of the occurrence of the other event. This concept is also known as cause and effect.
For the given ranks for two judges the difference between their ranks is:
d: 0.0 4.0 -2.0 1.0 -0.5 1.5 -1.0 -1.0 0.5 2.0
Squaring the given distance we have:
d²: 0.0 16.0 4.0 1.0 0.25 2.25 1.0 1.0 0.25 4.0
Σd² = 29.75
The Spearman's rank correlation coefficient is given as:
ρ = 1 - (6Σd²)/(n(n²-1))
ρ = 1 - (629.75)/(10(10²-1))
ρ ≈ 0.03
Since the correlation coefficient is close to zero, we can conclude that the two judges do not appear to agree in their standards.
Learn more about correlation here:
https://brainly.com/question/18498382
#SPJ1
The complete question is:
-6(4p+5) > 34-8p HELP ASAP
Answer:
p < -4
Step-by-step explanation:
-6(4p+5) > 34 - 8p
-24p - 30 > 34 - 8p
-16p - 30 > 34
-16p > 64
p < -4
sonny's select the options that will create the correct equation of the estimated population regression line.
The equation of the estimated population regression line is typically written in the form:
y = a + bx
where:
y is the dependent variable (or response variable)
x is the independent variable (or predictor variable)
a is the intercept (the value of y when x = 0)
b is the slope (the change in y for a unit change in x)
To create the correct equation of the estimated population regression line, Sonny should select the following options:
The variable y should be the dependent variable (or response variable).
The variable x should be the independent variable (or predictor variable).
The estimated intercept value should be plugged into the equation for a.
The estimated slope value should be plugged into the equation for b.
So, the correct equation of the estimated population regression line would be:
y = a + bx
where a and b are the estimated intercept and slope values, respectively.
To know more about Regression:
https://brainly.com/question/11805375
#SPJ4
(x^2-x-12)/(x+5)=x-6
There is no value of x that solves the proportion (x^2-x-12)/(x+5) = x-6.
How to solve the proportion?The proportion for this problem is defined by the equation presented as follows:
(x^2-x-12)/(x+5) = x-6.
As the measures are proportional, we can apply cross multiplication, hence:
x² - x - 12 = (x + 5)(x - 6)
x² - x - 12 = x² - x - 30
-12 = -30.
-12 = -30 is a false statement, hence there is no value of x which can solve the proportion (x^2-x-12)/(x+5)=x-6 presented in this problem.
Missing InformationThe problem asks for the value of x that solves the proportion (x^2-x-12)/(x+5) = x-6.
More can be learned about proportions at https://brainly.com/question/24372153
#SPJ1
PLEASE HELP !!!! HELP!!label each equation is proportionality or non proportional Help
y=9/x
y=x-12
h=3d
f=1/3e
Answer:
y=9/x => proportional
y = x - 12 ==> non-proportional
h = 3d ==> proportional
f = 1/3 e = proportional
Step-by-step explanation:
A proportional equation is of the general form
y = kx (directly proportional) or
y = k/x (inversely proportional)
k is known as the constant of proportionality
y = 9/x ==> k = 9 proportional
y = x - 12 cannot be expressed as y = kx or y = k/x
h = 3d ==> k = 3 proportional
f = 1/3 e ==> k = 1/3 proportional
Please answer the attached question
The values of e and f in the given equation are: e = 2√3 ± √(4√3), e = 2√3 ± 2√2, and f = 4√3.
How are radicals solved?Equations containing radicals can be made simpler by solving the resultant equation after squaring both sides of the equation to remove the radical. Nonetheless, caution must be exercised to guarantee that any solutions found are reliable and adhere to any variables' limitations.
The given equation is [tex](e - 2\sqrt{3} )^2[/tex] = f - 20√3.
Expanding the left side of the equation we have:
[tex](e - 2\sqrt{3} )^2[/tex] = (e - 2√3)(e - 2√3)
= [tex]e^2[/tex] - 2e√3 - 2e√3 + 12
= [tex]e^2[/tex] - 4e√3 + 12
Substituting back in the function
[tex]e^2[/tex] - 4e√3 + 12 = f - 20√3
[tex]e^2[/tex] - 4e√3 - f + 20√3 - 12 = 0
Using the quadratic formula:
e = [4√3 ± √(16*3 + 4(f - 20√3 + 12))] / 2
e = [4√3 ± √(4f - 64√3)] / 2
e = 2√3 ± √(f - 16√3)
Now for,
(e - 2√3)² = f - 20√3
(2√3 + √(f - 16√3) - 2√3)² = f - 20√3
f - 20√3 = f - 16√3
f = 4√3
Hence, the values of e and f in the given equation are: e = 2√3 ± √(4√3), e = 2√3 ± 2√2, and f = 4√3.
Learn more about radicals here:
brainly.com/question/1369233
#SPJ1