How many solutions does this equation have?

–20 − 5s = –5s + 6 − 12

The value of q is -27.

Recall Vieta's Formulas, which state that for a quadratic equation \(ax^2\) + bx + c = 0 with zeroes alpha and beta, the sum of the zeroes is equal to -b/a, and the product of the zeroes is equal to c/a.

In our equation \(x^2\) - 3x + q, the sum of the zeroes alpha and beta is -(-3)/1 = 3.

We are given that 2 alpha + 3 beta = 15. Substitute alpha = (15 - 3 beta)/2 into the equation.

Replace the value of alpha in the sum of zeroes equation: (15 - 3 beta)/2 + beta = 3.

Simplify the equation by multiplying both sides by 2: 15 - 3 beta + 2 beta = 6.

Combine like terms: 15 - beta = 6.

Subtract 15 from both sides: -beta = -9.

Multiply both sides by -1 to solve for beta: beta = 9.

Substitute the value of beta into the sum of zeroes equation: alpha = (15 - 3 * 9)/2 = -3.

Since we have the values of alpha and beta, we can find q using the product of the zeroes formula: q = alpha * beta = -3 * 9 = -27.

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Answer:

18 units to the right.

Step-by-step explanation:

Hope this helps!

The number of nonnegative integer solutions to the inequality x1 + x2 + x3 ≤ 11 is C(14,3) = 364.

We can solve this inequality by introducing an auxiliary variable x4, such that x1 + x2 + x3 + x4 = 11. Here, x1, x2, x3, and x4 are all nonnegative integers.

We can interpret this equation as follows: imagine we have 11 identical objects and we want to distribute them among four boxes (x1, x2, x3, and x4). Each box can contain any number of objects, including zero. The number of solutions to this equation will give us the number of nonnegative integer solutions to the original inequality.

We can use a technique known as stars and bars to count the number of solutions to this equation. Imagine we represent the 11 objects as stars: ***********.

We can then place three bars to divide the stars into four groups, each group representing one of the variables x1, x2, x3, and x4. For example, if we place the first bar after the first star, the second bar after the third star, and the third bar after the fifth star, we get the following arrangement:

| ** | * | ****

This arrangement corresponds to the solution x1=1, x2=2, x3=1, and x4=7. Notice that the number of stars to the left of the first bar gives the value of x1, the number of stars between the first and second bars gives the value of x2, and so on.

We can place the bars in any order, so we need to count the number of ways to arrange three bars among 14 positions (11 stars and 3 bars). This is equivalent to choosing 3 positions out of 14 to place the bars, which can be done in C(14,3) ways.

Therefore, the number of nonnegative integer solutions to the inequality x1 + x2 + x3 ≤ 11 is C(14,3) = 364.

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Answer:

1/2

Step-by-step explanation:

slope of perpendicular lines are always negative reciprocal of each other.

so slope of the given line = rise over run

that means -2/1 = -2

so slope of a line that is perpendicular to it will be negative of its reciprocal

that will be

\( \frac{1}{2} \)

The inverse of the matrix \(\left[\begin{array}{cc}2&-7\\-2&5\end{array}\right]\) is \(-\frac 14 \left[\begin{array}{cc}5&7\\2&2\end{array}\right]\)

From the question, we have the following parameters that can be used in our computation:

\(\left[\begin{array}{cc}2&-7\\-2&5\end{array}\right]\)

The determinant (D) of the matrix is

D = 2 * 5 + 7 * -2

Evaluate

D = -4

The inverse, I is then calculated as

\(I = -\frac 14 \left[\begin{array}{cc}5&7\\2&2\end{array}\right]\)

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The work done pulling the bucket to the top of the well is 104,299.4 pound-force-feet (lb*ft).

To find the work done pulling the bucket to the top of the well, we need to consider the weight of the bucket, the weight of the water, and the work done against gravity.

First, let's calculate the weight of the bucket and water combined. The weight of the bucket is 4 pounds, and the weight of the water is 35 pounds. Therefore, the total weight is 4 + 35 = 39 pounds.

Next, let's calculate the distance the bucket is pulled up. The well is 83 feet deep, so the bucket is pulled up a distance of 83 feet.

Now, let's calculate the work done against gravity. Work is calculated by multiplying the force applied by the distance over which the force is applied. In this case, the force is equal to the weight of the bucket and water, which is 39 pounds. The distance is 83 feet.

Work = Force * Distance
Work = 39 pounds * 83 feet

To calculate the work, we need to convert the weight from pounds to a unit called pound-force (lbf). 1 pound force is equal to the force exerted by a mass of 1 pound under acceleration due to gravity.

To convert from pounds to pound-force, we need to multiply by the acceleration due to gravity. The acceleration due to gravity is approximately 32.2 feet per second squared.

39 pounds * 32.2 feet per second squared = 1255.8 pound-force

Finally, we can calculate the work done pulling the bucket to the top of the well.

Work = 1255.8 pound-force * 83 feet
Work = 104,299.4 pound-force-feet (or lb*ft)

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Answer:

D=8

Step-by-step explanation:

normal division rules

33 goes into 47 1 time

then it leaves 17D

if we divide 170 by 33, we get 5 and if we divide 179 by 33, we still get 5. So we have 17D-165. That leaves us with something that you multiply 33 with to have a 2 in the ones place so that you get a remainder of 2. (In order to get 2 in the remainder, we need a 2 in the ones place because 4-2=2. In order to get that we need to see which number multiplied with 33 would get us a number smaller than 179 but also has a 2 in the ones place. Theres only one number and thats 4. 33*4=132. So in order to get only 2 in the remainder, we need the rest to be subtracted. This means that 17D-165=134. This way we can see that D=8

a)  (i)\(y' = (x^2 + 1)/(1 + x^2) + 2x * tan^{-1}(x) - 1\) (ii) y' = sinh(xlogx) * [1 + log(x)] b)\(y' = [x^2 * 4^{x^2} + cosh^6(3x)] * [2/x + 2xln(4) + 6cosh^5(3x) * sinh(3x) * 3]\) are the derivatives

(a) Finding the derivatives:

(i) To find the derivative of \(y = (x^2 + 1) tan^{-1}(x) - x,\) we can use the product rule and the chain rule. Let's calculate it step by step:

Using the product rule:

\(y' = [(x^2 + 1) * d(tan^{-1}(x))/dx] + [tan^{-1}(x) * d(x^2 + 1)/dx] - 1\)

Using the chain rule:

\(y' = [(x^2 + 1) * (1/(1 + x^2))] + [tan^{-1}(x) * 2x] - 1\)

Simplifying:

\(y' = (x^2 + 1)/(1 + x^2) + 2x * tan^{-1}(x) - 1\)

(ii) To find the derivative of y = sinh(xlogx), we can use the chain rule. Let's calculate it step by step:

Using the chain rule:

y' = d(sinh(xlogx))/dx * d(xlogx)/dx

Using the chain rule again for the second term:

y' = sinh(xlogx) * [d(xlogx)/dx]

Simplifying:

y' = sinh(xlogx) * [x * (1/x) + log(x)]

Simplifying further:

y' = sinh(xlogx) * [1 + log(x)]

(b) Using logarithmic differentiation, find y' if \(y = x^2 * 4^{x^2} + cosh^6(3x):\)

Let's take the natural logarithm of both sides of the equation:

\(ln(y) = ln(x^2 * 4^{x^2} + cosh^6(3x))\\Now, differentiate implicitly with respect to x:\\(d/dx) ln(y) = (d/dx) ln(x^2 * 4^{x^2} + cosh^6(3x))Using the logarithmic differentiation rule:\\y' / y = [(2x/x^2) + (d/dx)(x^2) * ln(4) + (d/dx)(cosh^6(3x))]Simplifying:\\y' / y = [2/x + 2xln(4) + 6cosh^5(3x) * (d/dx)(cosh(3x))]\\Using the chain rule for the last term:\\y' / y = [2/x + 2xln(4) + 6cosh^5(3x) * sinh(3x) * 3]Now, multiply both sides by y:\\\)

\(y' = y * [2/x + 2xln(4) + 6cosh^5(3x) * sinh(3x) * 3]Replace y with the original expression:\\\y' = [x^2 * 4^{x^2} + cosh^6(3x)] * [2/x + 2xln(4) + 6cosh^5(3x) * sinh(3x) * 3]\)

This is the derivative of y using logarithmic differentiation.

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Answer:

Equation of line is: \(\mathbf{y=4x+3}\)

Step-by-step explanation:

We are given:

A straight line has gradient 4 and passes through the point (5,23).

Work out the equation of the line.

The equation of line will be in slope-intercept form: \(y=mx+b\)

where m is slope and b is y-intercept

Finding slope

We have slope (gradient) m = 4

Finding y-intercept

Using slope m = 4 and point (5,23) we can find b i.e y-intercept

\(y=mx+b\\23=4(5)+b\\23=20+b\\b=23-20\\b=3\)

So, we get y-intercept b = 3

Finding Equation of Line

Now, writing equation of line having slope m = 4 and y-intercept b = 3 is:

\(y=mx+b\\y=4x+3\)

So, the Equation of line is: \(\mathbf{y=4x+3}\)

Answer:

First Option

Step-by-step explanation:

First, we will find the equation for the total cost of the tickets.

Given x = Cost of 1 adult ticket

y = Cost of 1 Child ticket

Total Cost of tickets at $40:

2x + 4y = 40

To find which graph represent the cost of the tickets,

we can substitute x = 0 to find y and vice-versa.

If x = 0 ,

\(2(0) + 4y = 40\\4y = 40\\y = \frac{40}{4} \\= 10\)

So when x = 0 , y = 10 which gives the coordinates (0,10).

Now if y = 0,

\(2x+4(0) = 40\\2x = 40\\x = \frac{40}{2} \\= 20\)

So when y = 0 , x = 20 which gives the coordinates (20,0)

This shows that this particular line passes through these 2 points and therefore the answer is the first option.

Answer: A i just took the test

Step-by-step explanation:

The two-dimensional cross-sections that can be obtained from a right rectangular prism is a rectangle.

A prism is a 3-dimensional shape with two identical shapes called bases facing each other. The other faces of a prism are parallelograms or rectangles.

The two-dimensional cross-sections that can be obtained from a right rectangular prism is a rectangle.

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Answer: 2/125

Explanation:

I hope this helped!

<!> Brainliest is appreciated! <!>

- Zack Slocum

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Answer:

16/1,000

Understanding place values:

The first 0 before the decimal point in 0.016 represents ones, the 0 after the decimal point represents tenths, the 1 represents hundredths and the 6 represents thousandths. We use 1,000 in our fraction because of the place value.

Step-by-step explanation:

1. Convert 0.016 into a fraction:

   0.016 = 16/1,000

2. Simpilfy the fraction 16/1,000:

   16/1,000 ÷ 8 = 2/125

2/125 is equivalent to 0.016

Side Note: Step 2. is only for people looking for the simplified answer!

            ~Hope this helps!

           ~Hocus Pocus

Answer:

erm

Step-by-step explanation:

The sides will be:

6cm

6cm

4cm

Answer:

In isosceles triangle two sides are equal . Let the length be x .hence perimeter can be given as x + x + 6 . Therefore 2x + 6 = 16 . And 2x = 10 .and hence x =5 . So the length of each side is 5 cm

Step-by-step explanation:

The recursive formula for the given problem is W(n) = W(n-1) + 2 * W(n-1), with the base case W(0) = 1. This formula calculates the number of n-letter "words" that can be created from an unlimited supply of 'a's, 'b's, and 'c's,

To derive the recursive formula, we consider two cases for the first letter of the word: either it is an 'a' or it is not. If the first letter is 'a', we need to ensure that the remaining (n-1) letters form a word with an even number of 'a's. Therefore, the number of words in this case is equal to W(n-1), as we are recursively solving for the remaining letters.

If the first letter is not 'a', we have the freedom to ch

oose from 'b' or 'c'. In this case, we have two options for each of the remaining (n-1) letters, resulting in 2 * W(n-1) possibilities. By summing these two cases, we obtain the recursive formula W(n) = W(n-1) + 2 * W(n-1), which calculates the total number of n-letter words satisfying the given criteria.

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a) P(30 ≤ x ≤ 32) = 0.3446.

b) P(30 ≤ x ≤ 32) = 0.2868.

c) The probability of getting values between 30 and 32 for x is higher in part (b) than in part (a).

We have,

(a)

The mean of the distribution is = 30 and the standard deviation is = 28.

For a sample size n = 31, the sample mean x follows a normal distribution with mean = 30 and standard deviation = /√n = 28/√31 = 5.02 (approx.).

Therefore, x ~ N(30, 5.02).

The probability P(30 ≤ x ≤ 32) can be found by standardizing the values using the formula z = (x - ) / , where z is the standard normal variable.

z1 = (30 - 30) / 5.02 = 0

z2 = (32 - 30) / 5.02 = 0.40

P(30 ≤ x ≤ 32) = P(0 ≤ z ≤ 0.40) = 0.3446 (approx.)

Therefore, x = 30, x = 5.02, and P(30 ≤ x ≤ 32) = 0.3446 (approx.).

(b)

For a sample size n = 62, the sample mean x follows a normal distribution with mean = 30 and standard deviation = /√n = 28/√62 = 3.56 (approx.).

Therefore, x ~ N(30, 3.56).

The probability P(30 ≤ x ≤ 32) can be found using the same method as in part (a).

z1 = (30 - 30) / 3.56 = 0

z2 = (32 - 30) / 3.56 = 0.56

P(30 ≤ x ≤ 32) = P(0 ≤ z ≤ 0.56) = 0.2868 (approx.)

Therefore, x = 30, x = 3.56, and P(30 ≤ x ≤ 32) = 0.2868 (approx.).

(c)

The standard deviation of part (b) is smaller than part (a) because of the larger sample size.

Therefore, the distribution about x is narrower in part (b) than in part (a). This means that the sample mean x in part (b) is likely to be closer to the population mean than the sample mean x in part (a).

As a result, the probability of getting values between 30 and 32 for x is higher in part (b) than in part (a).

Thus,

a) P(30 ≤ x ≤ 32) = 0.3446.

b) P(30 ≤ x ≤ 32) = 0.2868.

c) The probability of getting values between 30 and 32 for x is higher in part (b) than in part (a).

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Answer: $51.48

Step-by-step explanation:

If a $52 item is marked up by 10%, its new price will be:

$52 + 10% of $52 = $52 + $5.20 = $57.20

Then, if this new price is marked down by 10%, the final price will be:

$57.20 - 10% of $57.20 = $57.20 - $5.72 = $51.48

Therefore, the final price of the item after the markup and markdown is $51.48.

The combination of buses that can transport the 200 students at the lowest possible cost using no more than 8 drivers is 3 large buses and 2 small buses, at a total cost of $3,600.

Let's start by using just large buses. We would need 4 buses to transport all 200 students, at a cost of $3,200 (4 buses x $800 per bus). However, this would require 4 drivers, leaving only 4 drivers available for any additional buses.

Next, let's try using just small buses. We would need 5 buses to transport all 200 students, at a cost of $3,000 (5 buses x $600 per bus). This would also require 5 drivers, leaving only 3 drivers available for any additional buses.

Now, let's try a combination of large and small buses. Let's start with 3 large buses and 1 small bus. This would transport 190 students (3 buses x 50 seats + 1 bus x 40 seats), leaving 10 students who would need to be transported on another small bus. The total cost for this combination would be $3,000 (3 large buses x $800 per bus + 1 small bus x $600 per bus).

We still have 7 drivers remaining, so let's add another small bus to transport the remaining 10 students. This would bring the total cost to $3,600 (3 large buses x $800 per bus + 2 small buses x $600 per bus).

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Translate Question: A school must transport 200 students to an event. Both large and small buses are available. A large bus holds 50 people and costs $800 to rent for the event. A small bus holds 40 people and costs $600 to rent for the event. There are 8 drivers available on the day of the event. *

Find the combination of buses that can transport the 200 students at the lowest possible cost using no more than 8 drivers. •

Thus,  the average cost per unit for an output of q units is given by the equation c/q = 12000/q + 7, where the fixed cost is $12,000 and the variable cost is given by the function cv = 7q.

The given equation for the total cost of producing q units of a product is c = 5000 + 6q.

To find the average cost per unit for an output of q units, we need to divide the total cost by the number of units produced.

Thus, the average cost per unit can be written as c/q.

Substituting the given equation for c in terms of q, we get

c/q = (5000 + 6q)/q.

Simplifying this expression, we get c/q = 5000/q + 6.

Now, we are given that the fixed cost is $12,000 and the variable cost is given by the function cv = 7q.

The total cost equation c can be written as the sum of the fixed cost and the variable cost, i.e., c = 12000 + cv. Substituting the given equation for cv, we get c = 12000 + 7q.

Substituting this equation for c in terms of q in the expression we derived earlier for c/q, we get c/q = (12000 + 7q)/q. Simplifying this expression, we get c/q = 12000/q + 7.

Therefore, the average cost per unit for an output of q units is given by the equation c/q = 12000/q + 7, where the fixed cost is $12,000 and the variable cost is given by the function cv = 7q.

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Answer:

Step-by-step explanation:

To calculate the total surface area, you first need to calculate the surface area of one of the 4 triangles and multiply that by 4.

We can solve this two ways (I would recommend using 2).

1)

Using herons formula:

a, b, c = different sides of the triangle (length)

s = (a + b + c) / 2

a = \(\sqrt{s(s-a)(s-b)(s-c)}\)

If we input values:

a = b = c = 6.5m

s = (3 * 6.5) / 2 = 9.75

\(\sqrt{s(s-6.5)^{3} }\) which is about 18.29478 (exact answer: \(3.25\sqrt{31.6875}\))

18.29478 * 4 = the entire area = 73.17912 m^2

exact whole area: \((13\sqrt{31.6875})m^{2}\)

2)

Each of the four triangles can be divided into 2 more triangles.

(2 * 4 = 8 = total amount of new small triangles)

How we calculate the area of one of the new small ones:

(h * w) / 2 = a

h = 5.6m

w = (width of one of the four triangles width divided by 2) = 3.25m  

3.25m * 5.6m = 18.2m^2

18.2m^2 / 2 = 9.1m^2

There are 8 of the new small triangles in the biggest triangle made up of four smaller ones.

So if we take 9.1m^2 * 8, we get the total area!

Total area: 72.8m^2

Happy to help you!

If you have any questions, ask in the comments below!

Have a good day! (●'◡'●)

Answer:

2 terms

Step-by-step explanation:

terms include the coefficient and the variables

Answer:

2

Step-by-step explanation:

5a can be considered a term with a variable and 3 can be considered constant term, therefore 2 terms

hopefully this answer helped you!!!

Answer:

119 is the value of x when y = 7

Step-by-step explanation:

Since y varies inversely with x, we can use the following equation to model this:

y = k/x, where

Step 1:  Find k by plugging in values:

Before we can find the value of x when y = k, we'll first need to find k, the constant of proportionality.  We can find k by plugging in 49 for y and 17 for x:

Plugging in the values in the inverse variation equation gives us:

49 = k/17

Solve for k by multiplying both sides by 17:

(49 = k / 17) * 17

833 = k

Thus, the constant of proportionality (k) is 833.

Step 2:  Find x when y = k by plugging in 7 for y and 833 for k in the inverse variation equation:

Plugging in the values in the inverse variation gives us:

7 = 833/x

Multiplying both sides by x gives us:

(7 = 833/x) * x

7x = 833

Dividing both sides by 7 gives us:

(7x = 833) / 7

x = 119

Thus, 119 is the value of x when y = 7.

Answer:

x= 6.02953586.......

Step-by-step explanation:

Answer:

B'(-3, 2)

Step-by-step explanation:

The rule for a reflection over the y-axis is (x, y) → (-x, y)

This means that the x-values change while the y-values stay the same.

B(x, y) → (-x, y)

B(3, 2) → (-3, 2)

B'(-3, 2)

Hope this helps!

The volume of the cube given the dimensions of the side length is \(64x^{6} + 144x^{4} + 108x^{2} + 27\)

A cube is a 3-dimensional object that has six faces, twelve edges and eight vertices.

Volume of a cube = a³

Where a = length of a side

(4x² + 3) × (4x² + 3) = 16x^4 + 24x² + 9

(16x^4 + 24x² + 9) × (4x² + 3) = \(64x^{6} + 144x^{4} + 108x^{2} + 27\)

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The 95% confidence interval of voters not favoring the incumbent is (0.0706, 0.1294).

Sample size, n=400

Sample proportion, p = 40 / 400

= 0.1

We use normal approximation, for this, we check that both np and n(1-p) >5.

Since n*p = 40 > 5 and n*(1-p) = 360 > 5, we can take binomial random variable as normally distributed, with mean = p = 0.1 and standard deviation  = root( p * (1-p) /n )

= 0.015

For constructing Confidence interval,

Margin of Error (ME) =  z x SD = 0.0294

95% confidence interval is given by Sample Mean +/- (Margin of Error)

0.1 +/- 0.0294 = (0.0706 , 0.1294)

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Answer:

thanks

Step-by-step explanation:

Answer:

Thank you so much !

You're the best :)

Answer:

A line that slants down from left to right has a negative slope. Horizontal and vertical lines also have a slope.

Step-by-step explanation:

Based on its degree, and the number of terms, the polynomial is named as: bi-quadratic binomial.

A binomial is a type of polynomial that has two unrelated terms or unlike terms, having subtraction or addition sign separating the two terms. For example, the polynomial, ax² - b, has two unlike terms, ax² and b, and is therefore called a polynomial.

A bi-quadratic polynomial can be defined as a type of polynomial that has the term with the highest degree as 4. For example, the polynomial, ax^4 - b, is a bi-quadratic polynomial because the term that has the highest degree as 4 is ax^4.

Given the polynomial,  x^4 + 2:

There are two terms, x^4 and 2

The highest degree is 4.

Therefore, based on its degree, the polynomial is named as:  a bi-quadratic polynomial.

Based on the number of terms, the polynomial is named as: a binomial polynomial.

It can be named as: bi-quadratic binomial.

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