What is the volume of a cylinder with a radius of 3 feet and a height of 4 feet? Use 3.14 for pi. Round to the nearest hundredth.

The augmented matrix of a system of linear equations is given: , determine value(s) of k if 2 k - 2 if  the system has no solution there are no values of k that will make the system inconsistent.  The system has one solution for all values of k except k = 4.The system has infinitely many solutions if k = 0, a unique solution if k ≠ 0 and k ≠ 4, and no solutions if k = 4

To determine the value(s) of k for each case, we will perform row reduction on the augmented matrix and analyze the resulting echelon form.

1 3 1 | 0

2 k - 2 | 0

R2 - 2R1 -> R2

1 3 1 | 0

0 k - 4 | 0

Case (a): If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions. If k ≠ 4, then we can use back-substitution to find the solution(s):

k - 4 = 0 => k = 4

Since this contradicts our assumption, there are no values of k that will make the system inconsistent.

Case (b): If k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions.

Case (c): If k = 0, then the second row reduces to 0 = 0, which means we have a free variable and infinitely many solutions. If k ≠ 0 and k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the system is inconsistent with no solutions.

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Answer: 3rd option

Step-by-step explanation:

5500        55

-------- = ----------  

   p          100

After building 18 shelves and 24 tables, the carpenter will have 9 boxes of screws, 80 2x4s, and 39 sheets of plywood leftover.

To find the leftover materials, we need to calculate the total materials used to build 18 shelves and 24 tables, and then subtract that from the total materials the carpenter had on hand.

For 18 shelves, the carpenter used 18 boxes of screws, 36 2x4s, and 72 sheets of plywood. For 24 tables, the carpenter used 48 boxes of screws, 48 2x4s, and 72 sheets of plywood.

So, the carpenter used 66 boxes of screws, 84 2x4s, and 144 sheets of plywood in total.

Subtracting this from the materials on hand, we get 9 boxes of screws, 80 2x4s, and 39 sheets of plywood leftover.

Therefore, the carpenter can potentially use these leftover materials for future projects or sell them to recoup some of their costs.

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Answer:

the ans is f(x)=1.7x+21,472

The equation of the function is exponential and the function is f(x) = 21472(1.017)ˣ.

The population of a small town in Connecticut is 21,472 and the expected population growth is 1.7% each year.

Let's use a function to represent the town's population x years from now.

Hence,

1.7% = 1.7  / 100 = 0.017

Therefore,

f(x) = 21472(1 + 0.017)ˣ

Hence,

f(x) = 21472(1.017)ˣ

Therefore, the function is exponential.

The equation of the function is f(x) = 21472(1.017)ˣ

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The polynomial P(x) must have a degree of 4.

This is because a polynomial with real coefficients must have complex zeros in conjugate pairs. This means that if 1+i is a zero of the polynomial, then its conjugate, 1-i, must also be a zero. Similarly, if 2-i is a zero, then its conjugate, 2+i, must also be a zero. Therefore, the polynomial P(x) must have zeros at 1, 1+i, 1-i, 2-i, and 2+i. Since a polynomial's degree is equal to the number of its zeros, the polynomial must have a degree of 4.

In summary, a polynomial with real coefficients and zeros at 1, 1+i, and 2-i must have a degree of 4.

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The two possible solutions for the given triangle are:

Case 1: $B = 51.92°$, $C = 107.08°$, $c = 25.93$

Case 2: $B = 128.08°$, $C = 30.92°$, $c = 13.45$

Both solutions exist and are rounded to two decimal places.

The Law of Sines states that for any triangle ABC, the following equation holds:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Using this equation, we can solve for the missing angles and side lengths in the given triangle.

Case 1:
To find angle B, we can rearrange the equation to get:

$\sin B = \frac{b \sin A}{a}$

Plugging in the given values:

$\sin B = \frac{22 \sin 21°}{9.5}$

$\sin B = 0.789$

Taking the inverse sine of both sides:

$B = \sin^{-1}(0.789)$

$B = 51.92°$

To find angle C, we can use the fact that the sum of the angles in a triangle is 180°:

$C = 180° - A - B$

$C = 180° - 21° - 51.92°$

$C = 107.08°$

Finally, to find side c, we can use the Law of Sines again:

$\frac{c}{\sin C} = \frac{a}{\sin A}$

Rearranging and plugging in the given values:

$c = \frac{a \sin C}{\sin A}$

$c = \frac{9.5 \sin 107.08°}{\sin 21°}$

$c = 25.93$

So the solution for Case 1 is:

$B = 51.92°$, $C = 107.08°$, $c = 25.93$

Case 2:
In this case, we need to consider the possibility of an obtuse angle B. To find this angle, we can use the fact that the sine of an obtuse angle is the same as the sine of its supplement:

$\sin B = \sin (180° - B)$

So we can find the supplement of the angle we found in Case 1:

$B = 180° - 51.92°$

$B = 128.08°$

Plugging this value back into the Law of Sines equation, we can find the other missing values:

$C = 180° - A - B$

$C = 180° - 21° - 128.08°$

$C = 30.92°$

$c = \frac{a \sin C}{\sin A}$

$c = \frac{9.5 \sin 30.92°}{\sin 21°}$

$c = 13.45$

So the solution for Case 2 is:

$B = 128.08°$, $C = 30.92°$, $c = 13.45$

Therefore, the two possible solutions for the given triangle are:

Case 1: $B = 51.92°$, $C = 107.08°$, $c = 25.93$

Case 2: $B = 128.08°$, $C = 30.92°$, $c = 13.45$

Both solutions exist and are rounded to two decimal places.

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To make the remainder 0, the final value of k must be -1.

The question asks to find all values of k so that when x^(3)-k^(2)x+k+2 is divided by x-2, the remainder is 0.

To find the values of k, we need to use synthetic division.

First, we can write the equation as follows:

Now we can continue with the synthetic division:

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The equation for the line that passes through the point (4, -4) and is parallel to the line with the equation -6x - 2y = 14 is y = -3x + 8.

To find the equation for the line that passes through the point (4, -4) and is parallel to the line with the equation -6x - 2y = 14, we first need to find the slope of the given line. We can do this by rearranging the equation to solve for y and putting it in slope-intercept form, y = mx + b.

-6x - 2y = 14

-2y = 6x + 14

y = -3x - 7

The slope of the given line is -3. Since the line we are trying to find is parallel to the given line, it will have the same slope. Therefore, the slope of the line we are trying to find is also -3.

Now, we can use the point-slope form of a linear equation, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line, to find the equation of the line. Plugging in the slope and the point (4, -4), we get:

y - (-4) = -3(x - 4)

y + 4 = -3x + 12

y = -3x + 8

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The functions involved in g(x) are multiply by 2 and subtract the value of 5.

A function takes in values, applies specific operations to them, and produces an output. The inverse function acts, agrees with the outcome, and returns to the initial function. The graph of the inverse of a function shows the function and the inverse of the function, which are both plotted on the line y = x. This graph's line traverses the origin and has a slope value of 1.

Given that, f(x) involves the following functions:

Divide by 2

Add 5

Also, g(x) is the inverse of the function of f(x) hence, the function involved are inverse of the original function.

Hence, the functions involved in g(x) are multiply by 2 and subtract the value of 5.

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The value of −tanx is 1−secx, which is option c) in the given choices.

The correct answer is option c) 1−secx.

To find the value of −tanx, we can use the identity tanx = sinx/cosx. Multiplying both sides of the equation by −1 gives us −tanx = −sinx/cosx.

We can then substitute the value of cosx from the given equation into the equation for −tanx:

−tanx = −sinx/(1−sinx)

Multiplying both sides of the equation by (1−sinx) gives us:

−tanx(1−sinx) = −sinx

Distributing the −tanx on the left side of the equation gives us:

−tanx + tanx*sinx = −sinx

Rearranging the equation and factoring out sinx gives us:

tanx*sinx + sinx = tanx

sinx(tanx + 1) = tanx

Dividing both sides of the equation by (tanx + 1) gives us:

sinx = tanx/(tanx + 1)

Using the identity 1/cosx = secx, we can substitute secx for 1/cosx in the equation:

sinx = (sinx/cosx)/(sinx/cosx + 1/cosx)

Simplifying the equation gives us:

sinx = sinx/(sinx + secx)

Cross multiplying and rearranging the equation gives us:

sinx*(sinx + secx) = sinx

sinx^2 + sinx*secx = sinx

Subtracting sinx from both sides of the equation gives us:

sinx^2 + sinx*secx - sinx = 0

Factoring out sinx gives us:

sinx(sinx + secx - 1) = 0

Setting each factor equal to 0 gives us:

sinx = 0 or sinx + secx - 1 = 0

Solving for secx in the second equation gives us:

secx = 1 - sinx

Therefore, the value of −tanx is 1−secx, which is option c) in the given choices.

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a) maximum profit 20.5.

b) maximum profit 20,025.

c)  price 10.35

The given function, f(p)=-50p+2050p-20,700, is not a degree 3 polynomial. It is a degree 1 polynomial or a linear function. Therefore, the given conditions of degree 3 polynomial with integer coefficients and zeros 8i and 6/5 do not apply to this function.

Instead, we can use the given function to answer the questions about the company's monthly profit.

(a) To find the price that generates the maximum profit, we can use the formula for the vertex of a parabola, which is (-b/2a, f(-b/2a)). In this case, a = -50 and b = 2050.
The price that generates the maximum profit is -b/2a = -2050/(2*-50) = 20.5.

(b) To find the maximum profit, we can plug the price that generates the maximum profit into the function.
f(20.5) = -50(20.5) + 2050(20.5) - 20,700 = 20,025.

(c) To find the price(s) that would enable the company to break even, we can set the function equal to 0 and solve for p.
0 = -50p + 2050p - 20,700
20,700 = 2000p
p = 10.35

Therefore, the price that would enable the company to break even is 10.35. There is only one price that would enable the company to break even.

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a) At the time the article was published, the federal minimum wage was $5.85 per hour.

b) According to the US Department of Labor website, the federal minimum wage in 2008 was $6.55 per hour.  

c) To earn the current federal minimum wage of $7.25 per hour, a person must pick up a penny in 4.97 seconds.

d) The phrase "penny dreadful" is a British expression referring to cheaply printed stories of sensational and sometimes gruesome content that were sold for a penny in the 19th century.

A) The minimum wage when Owen wrote his article can be calculated by using the equation:

Minimum wage = (Amount of money earned)/(Amount of time taken to earn it)

In this case, the amount of money earned is $0.01 (the value of a penny) and the amount of time taken to earn it is 6.15 seconds. Therefore:

Minimum wage = ($0.01)/(6.15 seconds) = $0.00162601626 per second

To convert this to an hourly wage, we can multiply by the number of seconds in an hour (3600):

Minimum wage = ($0.00162601626 per second) x (3600 seconds per hour) = $5.85365854 per hour

Minimum wage =  $5.85

B) According to the U.S. Department of Labor, the federal minimum wage in 2008 was $6.55 per hour.

Therefore, Owen's arithmetic was slightly off, as his calculated minimum wage is lower than the actual minimum wage at that time.

C) To calculate how much time you would need to spend picking up a penny in order to earn the current minimum wage, we can use the same equation as before, but rearrange it to solve for the amount of time:

Amount of time = (Amount of money earned)/(Minimum wage)

The current federal minimum wage is $7.25 per hour, or $0.00201388889 per second. Therefore:

Amount of time = ($0.01)/($0.00201388889 per second) = 4.97 seconds

So you would need to spend 4.97 seconds picking up a penny in order to earn the current minimum wage.

D) The phrase "penny dreadful" refers to a type of cheap, sensationalist fiction that was popular in the 19th century. These stories were typically published in weekly installments and sold for a penny each, hence the name "penny dreadful."

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Answer:

5 compact discs on sale.

Step-by-step explanation:

To solve this you'll need to set up a system of equations. Let's use x or the original price and y for sale price. Here's what your equations will look like:

14.95x + 12.95y = 139.50

x + y = 10

Now, cancel out x so you can solve for y. You can choose either variable, but canceling out x gets you to your answer in less steps. Remember to multiply by a negative so you can cancel out the variable.

Here's what your work will look like:

14.95 + 12.95y = 139.50

-14.95 (x + y = 10)

Here's what your new equations will look like after distributing:

14.95x + 12.95y = 139.50

-14.95x - 14.95y = -149.5

Now, add these two equations together. When you do that, x cancels out and you can solve for y.

Here's what your new equation will look like after adding:

-2y = -10

Now, divide both sides by -2. After doing so, you should get y = 5. This means that you bought 5 compact discs on sale.

Hope this helps!

The effective annual IRR (internal rate of return) is 8%. If the dollars invested and withdrawn and the inflation rate for the 19-year time span of the investment is 9% per year the inflation-free IRR is -0.92%.

To calculate this, we need to use the IRR formula: IRR = [Sum of cash flows / (-initial investment)]1/n - 1, where n is the number of periods.

a. To find the effective annual IRR earned on this investment, we can use the following formula:

0 = -10,000/(1+IRR) - 10,000/(1+IRR)^2 - 10,000/(1+IRR)^3 - 10,000/(1+IRR)^4 + 10,000/(1+IRR)^5 + 10,000(1-0.05)/(1+IRR)^6       + 10,000(1-0.05)^2/(1+IRR)^7 + ... + 10,000(1-0.05)^14/(1+IRR)^18

The effective annual IRR earned on this investment to the nearest percent is 8%.

b. To find the inflation-free IRR earned on this investment, we can use the following formula:

Inflation-free IRR = (1 + IRR)/(1 + inflation rate) - 1

Plugging in the values we found in part (a), we get:

Inflation-free IRR = (1 + 0.08)/(1 + 0.09) - 1 = -0.0092

So the inflation-free IRR earned on this investment is approximately -0.92%.

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gf(x) = (x-5)(7x)

Part 1
a) The domain of f is all real numbers, since 7x is defined for all real numbers x. The domain of g is all real numbers, since x-5 is defined for all real numbers x. The domain of f+g is all real numbers, since the sum of two real numbers is a real number. The domain of f-g is all real numbers, since the difference of two real numbers is a real number. The domain of fg is all real numbers for which the product 7x(x-5) is defined, which is all real numbers except for x=5. The domain of ff is all real numbers, since the product of two real numbers is a real number. The domain of fg is all real numbers, since the product of two real numbers is a real number. The domain of gf is all real numbers, since the product of two real numbers is a real number.

b) (f+g)(x) = 7x + x - 5 = 8x - 5
(f-g)(x) = 7x - x + 5 = 6x + 5
(fg)(x) = 7x(x-5)
(ff)(x) = (7x)^2
fg(x) = (7x)(x-5)
gf(x) = (x-5)(7x)

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This is because the x-coordinate of a point on the graph of y = x^2 - x is given by the value of x that satisfies the equation. Similarly, the x-coordinate of a point on the graph of y = 20 is constant and equal to some value c.

To find the x-coordinates of the points where the graphs of the equations y = x^2 - x and y = 20 intersect, we need to solve the system of equations:

y = x^2 - x

y = 20

Since both equations are equal to y, we can set them equal to each other:

x^2 - x = 20

Now, if we solve for x, we will get the x-coordinates of the points where the two graphs intersect.

Hence, the x-coordinates of the points of intersection are the solutions of the equation x^2 - x = 20.

This is because the x-coordinate of a point on the graph of y = x^2 - x is given by the value of x that satisfies the equation. Similarly, the x-coordinate of a point on the graph of y = 20 is constant and equal to some value c.

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The equation of the circle that has center (-1,2) and passes through(-6,6) is (x + 1)^2 + (y - 2)^2 = 41.

The equation of a circle with center (h,k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2. In this case, the center of the circle is (-1,2) so h = -1 and k = 2. We can use the given point (-6,6) to find the radius of the circle.

The distance between the center and the given point is the radius of the circle, and we can use the distance formula to find it:
r = √((-6 - (-1))^2 + (6 - 2)^2)
r = √((-5)^2 + (4)^2)
r = √(25 + 16)
r = √41

So the equation of the circle is:
(x - (-1))^2 + (y - 2)^2 = (√41)^2
(x + 1)^2 + (y - 2)^2 = 41

Therefore, the equation of the circle that has center(-1,2) and passes through(-6,6) is (x + 1)^2 + (y - 2)^2 = 41.

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Answer:

To approximate the monthly interest rate, we need to use the formula for the annual percentage rate (APR) that takes into account the effect of compounding:

APR = (1 + r/m)^m - 1

where r is the annual interest rate, m is the number of compounding periods per year, and APR is the annual percentage rate.

In this case, r = 11.1% and m = 12 (since there are 12 months in a year). We want to solve for the monthly interest rate, which we can call r_m:

r_m = (1 + r/12)^12 - 1

r_m = (1 + 0.111/12)^12 - 1

r_m = 0.009141

To convert this to a percentage, we multiply by 100:

r_m = 0.9141%

Therefore, the approximate monthly interest rate is 0.9141%, rounded to two decimal places.

Answer: [tex]\frac{2}{35}[/tex] hour

Step-by-step explanation:

      We will divide the time it takes it to do 7 tasks (⅖ hour) by the number of tasks it does in that time frame (7 tasks) to find the time per task.

[tex]\frac{2}{5}[/tex] hour / 7 tasks = [tex]\frac{2}{5} *\frac{1}{7}[/tex] = [tex]\frac{2}{35}[/tex] hour or about 3.43 minutes

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The requried rate of change of the function f[x] on the interval -5 ≤ x ≤ 0 is F(x)' = -11/5.

Rate of change is defined as the change in value with rest to the time is called rate of change.

Here,
From the graph we have,
F(-5) = 1 and F(0) = -10

So the rate of change is given as,
F(x)' = F(0) - F(-5) / 0 + 5
F(x)' = -10 - 1 / 5
F(x)' = -11/5

Thus, the requried rate of change of the function f[x] on the interval -5 ≤ x ≤ 0 is F(x)' = -11/5.

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Answer:

80 and 2x + 9

Step-by-step explanation:

to evaluate g(f(1)) , evaluate f(1) then substitute the value obtained into g(x)

f(1) = 3(1) + 7 = 3 + 7 = 10 , then

g(10) = 10² - 2(10) = 100 - 20 = 80

------------------------------------------------------

to calculate (g ￮ f)(x) , substitute x = f(x) into g(x)

(g ￮ f)(x)

= g(f(x))

= g(x + 4)

= 2(x + 4) + 1

= 2x + 8 + 1

= 2x + 9

According to the given information, the probabilites are a) 0.159, b)0.107, d) 0.893 e) 0.662 c) expected value is 12.192 f) standard deviation is 3.0121.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

a) Using the binomial probability formula, we have:

P(X = 13) = (48)^13 * (0.254)^13 * (0.746)^35

= 0.159

So the probability that 13 out of 48 flowers will bloom early is 0.159.

b) To find the probability that fewer than 13 flowers will bloom early, we can find the cumulative probability up to 12:

P(X < 13) = P(X = 0) + P(X = 1) + ... + P(X = 12)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=0 to 12

= 0.107

So the probability that fewer than 13 flowers will bloom early is 0.107.

c) The expected value of a binomial distribution is given by n*p, where n is the number of trials and p is the probability of success. In this case, we have:

E(X) = 48 * 0.254

= 12.192

So on average, we expect to observe about 12.192 early blooming flowers.

d) To find the probability that at least 13 flowers will bloom early, we can use the complement rule and find the probability that 12 or fewer flowers will bloom early, and subtract that from 1:

P(X ≥ 13) = 1 - P(X < 13)

= 1 - 0.107

= 0.893

So the probability that at least 13 flowers will bloom early is 0.893.

e) To find the probability that between 9 and 14 flowers (inclusive) will bloom early, we can find the cumulative probability from 9 to 14:

P(9 ≤ X ≤ 14) = P(X = 9) + P(X = 10) + ... + P(X = 14)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=9 to 14

= 0.662

So the probability that between 9 and 14 flowers (inclusive) will bloom early is 0.662.

f) The variance of a binomial distribution is given by np(1-p), and the standard deviation is the square root of the variance. In this case, we have:

Var(X) = 48 * 0.254 * (1-0.254)

= 9.078

SD(X) = sqrt(Var(X))

= sqrt(9.078)

= 3.0121 (rounded to 4 decimal places)

So the standard deviation of the number of flowers that bloom early in a row of 48 flowers is approximately 3.0121.

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Answer: 6 and 12

Step-by-step explanation:

The value of x in the function 4x+6=8x-22 is 7.

The slope is equal to -10.

The slope-intercept form of the function 4x+5y=24 is y = -4x/5 + 24/5.

The value of f(-5) is equal to -36.

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Substituting the given points into the slope formula, we have the following;

Slope, m = (-12 - 8)/(-4 + 6)

Slope, m = -20/2

Slope, m = -10.

4x+6=8x-22

8x - 4x = 22 + 6

4x = 28

x = 7.

4x + 5y = 24

5y = -4x + 24

y = -4x/5 + 24/5

For f(-5), we have:

f(x)=7x-1

f(-5)=7(-5)-1

f(-5) = -36.

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Answer:

13400

Step-by-step explanation:

A system of inequalities to represent the constraints of this problem are x ≥ 0 and y ≥ 0.

A graph of the system of inequalities is shown on the coordinate plane below.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of HD Big View television produced in one day and number of Mega Tele box television produced in one day respectively, and then translate the word problem into algebraic equation as follows:

Since the HD Big View television takes 2 person-hours to make and the Mega TeleBox takes 3 person-hours to make, a linear equation to describe this situation is given by:

2x + 3y = 192.

Additionally, TVs4U’s total manufacturing capacity is 72 televisions per day;

x + y = 72

For the constraints, we have the following system of linear inequalities:

x ≥ 0.

y ≥ 0.

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To find a general solution to the equation ax^(2) + bx + 1 = 0 using the completion of the square, we need to follow these steps:

1. Divide the entire equation by a to get x^(2) + (b/a)x + 1/a = 0.

2. Move the constant term to the right side of the equation: x^(2) + (b/a)x = -1/a.

3. Complete the square by adding (b/2a)^(2) to both sides of the equation: x^(2) + (b/a)x + (b/2a)^(2) = -1/a + (b/2a)^(2).

4. Factor the left side of the equation: (x + b/2a)^(2) = -1/a + (b/2a)^(2).

5. Take the square root of both sides of the equation: x + b/2a = ±√(-1/a + (b/2a)^(2)).

6. Solve for x: x = -b/2a ± √(-1/a + (b/2a)^(2)).

This is the general solution to the equation.

The conditions for both solutions to be real are that the discriminant, -1/a + (b/2a)^(2), is greater than or equal to 0. This means that b^(2) - 4a >= 0.

The conditions for both solutions to be complex numbers are that the discriminant, -1/a + (b/2a)^(2), is less than 0. This means that b^(2) - 4a < 0.

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Answer:

d=69, e=32, f=79

Step-by-step explanation:

e=32

d=69

f=180-69-32=79

The measure of each of the missing angles include the following:

d = 69.

e = 32.

f = 79.

In Geometry, the vertical angles theorem is also referred to as vertically opposite angles theorem and it states that two (2) opposite vertical angles that are formed whenever two (2) lines intersect each other are always congruent, which simply means being equal to each other.

By applying the vertical angles theorem to the geometric figure, we have the following:

m∠e = 32°

m∠d = 69°

From the linear postulate theorem, we have:

m∠e + m∠d + m∠f = 180°

32° + 69° + m∠f = 180°

m∠f = 180° - (32° + 69°)

m∠f = 79°

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The solution to the system of equations is (4/3, 6, -2).

To solve the system of linear equations using the Gauss-Jordan elimination method, we need to perform row operations to reduce the system to reduced row echelon form. This will allow us to easily solve for the variables. Here are the steps:

1. Start with the given system of equations:

{ 6x - 2y + 2z = 4
{ 3x - y + 2z = 2
{ -12x + 4y - 8z = 8

2. Write the system as an augmented matrix:

[ 6 -2 2 | 4 ]
[ 3 -1 2 | 2 ]
[ -12 4 -8 | 8 ]

3. Divide the first row by 6 to get a leading 1:

[ 1 -1/3 1/3 | 2/3 ]
[ 3 -1 2 | 2 ]
[ -12 4 -8 | 8 ]

4. Use the first row to eliminate the x terms in the second and third rows:

[ 1 -1/3 1/3 | 2/3 ]
[ 0 2/3 5/3 | 2/3 ]
[ 0 0 -4 | 8 ]

5. Divide the second row by 2/3 to get a leading 1:

[ 1 -1/3 1/3 | 2/3 ]
[ 0 1 5/2 | 1 ]
[ 0 0 -4 | 8 ]

6. Use the second row to eliminate the y terms in the first and third rows:

[ 1 0 7/6 | 5/6 ]
[ 0 1 5/2 | 1 ]
[ 0 0 -4 | 8 ]

7. Divide the third row by -4 to get a leading 1:

[ 1 0 7/6 | 5/6 ]
[ 0 1 5/2 | 1 ]
[ 0 0 1 | -2 ]

8. Use the third row to eliminate the z terms in the first and second rows:

[ 1 0 0 | 4/3 ]
[ 0 1 0 | 6 ]
[ 0 0 1 | -2 ]

9. The system is now in reduced row echelon form, and we can easily solve for the variables:

x = 4/3
y = 6
z = -2

So the solution to the system of equations is (4/3, 6, -2).

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