use the grid to create a model to solve the percent problem. what is 20% of 90? responses 12 12 14 14 16 16 18

To solve the equation 10x dy/dx - 2y = x-3, we will use the method of separation of variables, we get 5x ln(2y - x + 3) + (5/2) x^2 - (1/2) (2y - x + 3)^2 = x + C

This method involves rearranging the equation so that all terms involving x are on one side and all terms involving y are on the other side. Then, we can integrate both sides of the equation to find the general solution. Here are the steps:

Rearrange the equation to separate the variables:

10x dy/dx = 2y + x - 3

10x dy = (2y + x - 3) dx

Separate the variables:

10x dy = 2y dx + x dx - 3 dx

10x dy - 2y dx = x dx - 3 dx

(10x dy - 2y dx)/(2y - x + 3) = dx

Integrate both sides of the equation:

∫(10x dy - 2y dx)/(2y - x + 3) = ∫dx

Use the substitution method to solve the integral on the left side of the equation. Let u = 2y - x + 3, then du = 2 dy - dx:

∫(10x (du/2 + dx) - 2u du)/(u) = ∫dx

5x ∫du + 5 ∫x dx - ∫u du = ∫dx

5x ln(u) + (5/2) x^2 - (1/2) u^2 = x + C

Substitute back for u and simplify:

5x ln(2y - x + 3) + (5/2) x^2 - (1/2) (2y - x + 3)^2 = x + C

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The parametric equations for the line: x(t) = 6+t, y(t) = -7+3t and z(t) = 4 - 2/3t.

Given:The point (6,−7,4) The vector (1,3,−23)To find:A vector equation and parametric equation for the line.Using vector equation:We know that vector equation of line is given by :r= a+ t * bWhere,a is the vector which passes through given point (6,−7,4)and,b is the given vector parallel to the line..r(t) = (6, −7, 4) + t(1, 3, −2/3)r(t) = (6+t, -7+3t, 4 - 2/3t) Parametric Equation:

Parametric equations are the set of equation where each variable x,y and z are expressed in terms of a single variable t.

Substitute x,y and z in above vector equation.r(t) = (6+t, -7+3t, 4 - 2/3t)Therefore, parametric equations for the line: x(t) = 6+t, y(t) = -7+3t and z(t) = 4 - 2/3t

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The quotient of 4.6 x 10⁵ over 2,300,000 is equal to 0.002.

Quotient is the way in which a number is use to divide another number which gives us  the answer.

The quotient of 4.6 x 10⁵ over 2,300,000 represents the result of dividing the number 4.6 x 10⁵ by the number 2,300,000.

Firstly, we can write 4.6x10⁵ as 0.46 x 10⁶,

since 10⁵ is equal to 100,000 and 10⁶ is equal to 1,000,000. So, we can rewrite the expression as:

[tex]$\frac{0.4 \times 10^6} { 2,300,000}[/tex]

Next, we can simplify this expression by dividing both the numerator and denominator by 10⁶. This gives us:

0.46 / 2.3

Now, we can simplify this expression further by dividing both the numerator and denominator by 0.01, which is equivalent to multiplying by 100. This gives us:

0.46 / 2.3 = 46 / 23000 = 0.002

Therefore, the quotient of 4.6x10⁵ over 2,300,000 is equal to 0.002.

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To write an augmented matrix for a system of equations, you need to organize the equations into a matrix format.

This is done by writing each equation in terms of its coefficients, separating them with a comma, and placing them in a row. The coefficients for each equation should line up in the same columns. Then you need to include the constant terms of each equation in a separate column to the right of the coefficients.

For example, if you have the equations 2x + 3y = 8 and 5x + 7y = 10, the augmented matrix would be:

2, 3, 8
5, 7, 10

An augmented matrix for a system of equations is a matrix that represents the system of linear equations.

To write an augmented matrix for a system of equations, you can follow the following steps:

Step 1: Write the system of equations in matrix form.Example: Consider the following system of linear equations2x + 3y + 4z = 4x - 2y + 5z = 3x + y - z

Step 2: Write the matrix for the coefficients of the variables of the system of equations along with the matrix of the constant terms

Example: The matrix of the coefficients of the variables and the matrix of the constants for the above system of linear equations can be written as [2 3 4 | 4; 4 -2 5 | 3; 3 1 -1 | 2]

Step 3: Write the augmented matrix by combining the matrix of the coefficients and the matrix of the constants with a vertical line in between.

Example: The augmented matrix for the above system of linear equations is[2 3 4 | 4; 4 -2 5 | 3; 3 1 -1 | 2]

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As a result, each spot in W would be reduced in size as respect by a ratio of 0.5. (0, 0). Because of this, the symbol of W will be a condensed version of the initial collection of points that are fixed at the origin.

When b doesn't really equal 0, an ordered list of values a and b, represented as a / b, is said to be a ratio. Two ratios is set to be equal in an equation called a proportion. The ratio would be written as 1: 3 if there were 1 boy & 3 girls, for instance.

Using the next transformation will allow us to determine the picture of a group of points W following a dilatation with centre (0,0) and scale factor of 0.5:

The point inside the image W' that corresponds to a point in W with coordinates x, y is: x', y' = 0.5x, 0.5y

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The average rate of change for the volume of water recommended from 2 fish to 10 fish is equal to 2.5 gallons of water per fish.

Average rate of change from 2 fish to 10 fish can be calculated as follows,

Average rate of change = (change in y) / (change in x)

⇒ Average rate of change = (y₂ - y₁) / (x₂ - x₁)

where (x₁ y₁) = (2, 5) and (x₂, y₂) = (10, 25).

Substituting these values, we have,

⇒ Average rate of change = (25 - 5) / (10 - 2)

⇒ Average rate of change =  = 20 / 8

⇒ Average rate of change = = 2.5

This means that,

On average for each additional fish added to the aquarium between 2 and 10 fish,

The recommended volume of water increases by 2.5 gallons.

Therefore, the average rate of change from 2 fish to 10 fish is 2.5 gallons of water per fish.

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The above question is incomplete, the complete question is:

The table shows the volume of water y,

in gallons, recommended for x fish in an

aquarium.

x  :   2           4             6             8             10

y  :   5          10             15           20            25

What is the average rate of change

from 2 fish to 10 fish, and what does it mean?

Answer:

d(0) = -3

Step-by-step explanation:

d(0) = -0 + -3

= 0 + -3

= -3

The creation of strata, which are proportional to the size

What is Sampling?

Sampling refers to the process of selecting a subset of individuals or items from a larger population, in order to study and draw conclusions about the population. Sampling is often used in research, marketing, and other fields to collect data from a smaller group, which is then analyzed to make inferences or predictions about the larger population.

There are several different methods of sampling, including random sampling, stratified sampling, cluster sampling, and convenience sampling. Each method has its own strengths and weaknesses, and the choice of sampling method will depend on the research question, the size of the population, and other factors.

A sample is biassed when it does not accurately reflect the population that it is supposed to represent. A sample statistic (such the sample mean or proportion) that consistently overvalues or undervalues the real population parameter can result from this. The sample technique, sampling frame, and nonresponse are only a few examples of the variables that might lead to bias. In order for the sample to effectively represent the population and for valid statistical inferences to be formed, sampling bias must be minimised.

Hence, The creation of strata, which are proportional to the size.

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Answer:

1st answer choice

no because there is no whole number that when multiplied by itself gives 125

Step-by-step explanation:

125 is a perfect cube

5 times 5 times 5 = 125

Answer:

Option A

Step-by-step explanation:

125 is NOT a perfect square, because there is no whole number that multiplies by itself to give 125.

To check the perfectness of a square, take the square root:

[tex]\sqrt{125} =[/tex] 11.1803399

The above number is NOT a whole number

∴125 is NOT a perfect square

Since y = x + 13 you have only to replace x with values on the table:

Now we have the coordinates of the points (x, y):

Now draw the points and connect them. I hope this helped.

For the given numerical example in a classification problem, the logistic loss function is always less than or equal to zero, indicating that a solution that minimizes the objective function cannot be found, possibly due to non-linearly separable data.

Since this is a classification problem with binary labels, we can use the logistic loss function given by:

L(y, h(x)) = log(1 + exp(-y * h(x)))

where y is the true label (either 1 or -1), h(x) is the predicted value, and exp is the exponential function.

In this case, λ = 0.5, y = 1, x = [10], and θ^ = [θ1^,θ2^]. We want to minimize the objective function:

f(θ^) = λ/2 * ||θ^||^2 + L(y, θ^ · x)

Substituting in the values, we get:

f(θ^) = 0.25 * (θ1^2 + θ2^2) + log(1 + exp(-θ1^ * 10))

To solve for θ1^ and θ2^, we need to find the partial derivatives of f(θ^) with respect to θ1^ and θ2^ and set them equal to zero:

∂f(θ^)/∂θ1^ = 0.1 * exp(-θ1^ * 10) * (1 / (1 + exp(-θ1^ * 10))) + 0.5 * θ1^ = 0

∂f(θ^)/∂θ2^ = 0.5 * θ2^ = 0

The second equation gives θ2^ = 0. Setting the first equation to zero and simplifying, we get:

exp(-θ1^ * 10) = -5

Since the exponential function is always positive, there are no solutions to this equation. However, we can use the hint given in the problem to show that the loss function is always less than or equal to zero for this example:

L(y, θ^ · x) = log(1 + exp(-10 * θ1^))

= log(1 + exp(-10 * (θ1^ - log(5))))

Since exp(-10 * (θ1^ - log(5))) is always less than or equal to 1, log(1 + exp(-10 * (θ1^ - log(5)))) is always less than or equal to 0. Therefore, the loss function is always less than or equal to 0.

So, we cannot find a solution that minimizes the objective function for this example. This suggests that the data is Non -linearly separable.

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_____The given question is incomplete, the complete question is given below:

Consider minimizing the above objective fuction for the following numerical example:

λ=0.5,y=1,x=[10]

Note that this is a classification problem where points lie on a two dimensional space. Hence θ^ would be a two dimensional vector.

Let θ^=[θ1^,θ2^], where θ1^,θ2^ are the first and second components of θ^ respectively.

Solve for θ1^,θ2^.

Hint: For the above example, show that Lossh(y(θ^⋅x))≤0

Answer:

18

Step-by-step explanation:

if you rotate the shape and if the shapes are the same the answer is 18

Answer:

IT IS D 5.5 FEET!!!!!

Answer:

16 and -16

Step-by-step explanation:

The expected number of times it will land on pink in 4 spins is then (1/2) x 4 = 2.

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain. The probability of an event can be determined by dividing the number of favorable outcomes by the total number of possible outcomes. It is used in many fields, including mathematics, statistics, physics, engineering, finance, and more, to make predictions and inform decision-making.

Here,

Assuming that the spinner is fair (i.e., the probability of landing on each section is equal), the best prediction for the number of times it will land on pink when the spinner has 4 pink sections and 4 blue sections is 2. This is because there are a total of 8 sections, so the probability of landing on pink on any one spin is 4/8 or 1/2.

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Complete question:

If you spin the spinner 4 times when it has 4 pink sections and 4 blue sections, what is the best prediction possible for the number of times it will land on pink?

The graph of g(t) will be a straight line with a slope of ½ for 0 ≤ t < 6 and a horizontal line at g(t) = 3 for t ≥ 6.

To solve the initial value problem y'' + y = g(t), y(0) = 0, y'(0) = 1, where g(t) = t/2 for 0 ≤ t < 6 and g(t) = 3 for t ≥ 6, we first take the Laplace transform of both sides of the differential equation:

L{y'' + y} = L{g(t)}

Using the properties of the Laplace transform, we can rewrite the left-hand side as: L{y''} + L{y} = L{g(t)}

Now, we can use the initial conditions y(0) = 0 and y'(0) = 1 to find L{y''} and L{y}: L{y''} = s² Y(s) - s y(0) - y'(0) = s² Y(s) - 1 L{y} = Y(s)

Substituting these back into the equation, we get:

s² Y(s) - 1 + Y(s) = L{g(t)}

Next, we need to find the Laplace transform of the forcing function g(t).

Since g(t) is piecewise-defined, we can use the Heaviside step function to write it as:

g(t) = (t/2) + 3 H(t - 6)

Taking the Laplace transform of this expression gives us:

L{g(t)} = L{(t/2) + 3 H(t - 6)} = (1/2) L{t} + 3 L{H(t - 6)}

Using the properties of the Laplace transform, we can find L{t} and L{H(t - 6)}:

L{t} = 1/s² L{H(t - 6)} = e^(-6s)/s

Substituting these back into the equation for L{g(t)}, we get:

L{g(t)} = (½)(1/s²) + 3 (e^(-6s)/s)

Now, we can substitute this expression for L{g(t)} back into the equation for s² Y(s) - 1 + Y(s) = L{g(t)}:

s² Y(s) - 1 + Y(s) = (1/2)(1/s²) + 3 (e^(-6s)/s)

Solving for Y(s), we get: Y(s) = (1 + s²/2 + 3s e^(-6s))/(s² + s)

Finally, we can take the inverse Laplace transform of Y(s) to find the solution y(t): y(t) = L⁻¹{Y(s)} = L⁻¹{(1 + s²/2 + 3s e^(-6s))/(s² + s)}

To sketch the graphs of the forcing function g(t) versus t and the solution y(t) versus t, we can plot the expressions for g(t) and y(t) on a graph with t on the horizontal axis and g(t) or y(t) on the vertical axis.

The graph of g(t) will be a straight line with a slope of ½ for 0 ≤ t < 6 and a horizontal line at g(t) = 3 for t ≥ 6.

The graph of y(t) will be a curve that is determined by the expression for y(t) that we found using the Laplace transform.

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Answer:

Step-by-step explanation:

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the maturity value of the loan, P is the principal (the amount borrowed), r is the annual interest rate (12% in this case), n is the number of times the interest is compounded per year (12 for monthly compounding), and t is the time period in years (6 years in this case).

(a) To find the maturity value of the loan, we can substitute the given values into the formula and solve for A:

A = 9000(1 + 0.12/12)^(12 x 6)

A = 9000(1.01)^72

A = 18,137.60

Therefore, the maturity value of the loan is $18,137.60.

(b) To find the amount of interest paid on the loan, we can subtract the principal from the maturity value:

Interest = A - P

Interest = 18,137.60 - 9000

Interest = 8,137.60

Therefore, the amount of interest paid on the loan is $8,137.60.

The answer is (D) No; the function cannot be represented by a straight line.

What is the slope?

In mathematics, slope refers to the steepness or incline of a line on a graph. It is a measure of how much the dependent variable changes for every unit change in the independent variable.

To determine whether the function can be represented by a straight line, we need to check whether the slopes of all the segments between the points are equal.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

Using this formula, we can calculate the slopes of the segments:

slope of TU = (1 - 4) / (-2 - (-5)) = 1/3

slope of VW = (-4 - (-2)) / (1 - (-1)) = -1

slope of WT = (-2 - 4) / (1 - (-5)) = -3/2

slope of TV = (-2 - 4) / (-1 - (-5)) = 3/2

slope of UV = (1 - 4) / (-2 - (-5)) = 1/3

From these calculations, we can see that the slopes of the segments TU and UV are equal, and the slopes of segments VW and TV are equal, but the slopes of segments UV and TV are not equal.

Therefore, the answer is (D) No; the function cannot be represented by a straight line.

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The function g is given by g(x) =x2 + 3. The question is asking to find the values of g(x + h) - g(x), g(-4), g(0), g(8), g(a+h), and hg(-4).

The function g is given by g(x) =x² + 3. Therefore, g(x + h) - g(x) = (x + h)² + 3 - (x² + 3)= x² + 2hx + h² + 3 - x² - 3 = 2hx + h².

If x = −4, g(-4) = (-4)²+3= 19. Therefore, g(-4) = 19.

If x=0,  g(0)=0²+3=3. Therefore, g(0) = 3.

If x=8, g(8)=8²+3=67. Therefore, g(8)=67.

If x = a + h,  g(a + h) = (a + h)²+3 = a²+2ah+h²+3. Therefore,  g(a + h) = a²+2ah+h²+3.

Since we are given g(−4) = 19, so, hg(-4) = h(19) = 19h. Therefore, hg(-4) = 19h.

Upon simplifying g(-4 + h), we get : g(-4 + h) = h²-8h+19.

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The greatest common factor of [tex]6x^2 - 15[/tex] is 3. The greatest common factor among these numbers is 3, which is the biggest number by which any of them can be split.

Define common factors. Mathematicians refer to factors that are shared by two or more numbers as common factors. A common factor is a number that will split a collection of two or more numbers exactly, to put it another way. a list of definitions for common factors. a number that divides other numbers equally, or more. synonyms include common measure and common divisor. The term "Greatest Common Factor," often referred to as the "Greatest Common Divisor" (GCD) or the "Highest Common Factor" (HCF), is frequently shortened as "GCF."

[tex]6x^2 - 15=0[/tex]

Calculate.

[tex]6=3*2\\15=3*5[/tex]

The greatest number that can be used to divide both of the other numbers.

3

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Line segments are straight lines that have a beginning and an end point and do not intersect with each other. They lie on the same plane and have the same direction.

Line segments are straight lines with a beginning and an end point and they do not intersect with each other. They are usually used to represent a certain length, distance, or area of a given space. Line segments have the same direction and lie on the same plane. They can be used to measure distances between two points or to establish boundaries of certain locations. Line segments are also used to plot out shapes and angles in math and geometry. Line segments can be used to create patterns, create shapes, and create figures. They are also used to connect two points and form a line. Line segments can be used to determine the distance between two points, to draw a line between two points, and to create a boundary. Line segments can be used to identify an area or region, to measure the size of an object, and to plot out a pattern. In conclusion, line segments are straight lines that have a beginning and an end point and do not intersect with each other. They lie on the same plane and have the same direction.

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the new coordinates are: A' = (5, 4) B' = (6, 0) C' = (5, 8)

Why it is and what are coordinates?

To translate the triangle, we need to add the same values to each coordinate. Let's say we want to translate the triangle by adding the vector <2, 3> to each point.

A' = (3 + 2, 1 + 3) = (5, 4)

B' = (4 + 2, -3 + 3) = (6, 0)

C' = (3 + 2, 5 + 3) = (5, 8)

Therefore, the new coordinates are:

A' = (5, 4)

B' = (6, 0)

C' = (5, 8)

In geometry, coordinates are values that specify the position of a point or an object in a plane or in space. In a two-dimensional plane, a point can be located by its distance from the origin (0,0) along the x-axis (horizontal) and y-axis (vertical).

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The coordinates of the fourth vertex are (8,6). The solution has been obtained by using concept of slope.

What is a slope?

By dividing the change in the y coordinate by the change in the x coordinate, one can determine a line's slope in mathematics.



We are given vertices as A (-4,7), B (3,5) and C (1,-4).

Let the vertices of D be (x,y)

We know that opposite sides of a parallelogram are parallel.

So,

Slope of AB = Slope of CD

Now, using the formula, we get

⇒ [tex]\frac{5-7}{3+4}[/tex]  =  [tex]\frac{y+4}{x-1}[/tex]

⇒ [tex]\frac{-2}{7}[/tex]  =  [tex]\frac{y+4}{x-1}[/tex]

⇒-2 (x - 1) = 7 (y + 4)

⇒-2x + 2 = 7y + 28

⇒-2x = 7y + 26

⇒x = -3.5y - 13    ...(1)

Similarly,

Slope of AC = Slope of BD

So, using the formula, we get

⇒ [tex]\frac{-4-7}{1+4}[/tex] =  [tex]\frac{y-5}{x-3}[/tex]

⇒ [tex]\frac{-11}{5}[/tex] =  [tex]\frac{y-5}{x-3}[/tex]

⇒ -11 (x - 3) = 5 (y - 5)

⇒ -11x + 33 = 5y - 25

⇒ -11x = 5y - 58    ...(2)

On substituting (1) in (2), we get

⇒ -11 (-3.5y - 13) = 5y - 58

⇒ 38.5y + 143 = 5y - 58

⇒ 33.5y = -201

⇒ y = -6

So,

⇒x = -3.5 (-6) - 13

⇒x = 21 - 13

⇒x = 8

Hence, the coordinates of the fourth vertex are (8,6).

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Answer:

A rational function is undefined when the value of the denominator is 0, so:

[tex]x+2=0\\\therefore x=-2[/tex]

Thus, the correct answer is -2.

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

the length of the box is 5 centimeters.

How to solve?

Since the box is cube-shaped, all sides have the same length. Let's call the length of one side "x".

The volume of the cube can be calculated by raising the length of one side to the third power:

V = x²3

We know that the volume of the box is 125 cubic centimeters, so we can set up an equation:

125 = x²3

To solve for x, we can take the cube root of both sides:

∛125 = ∛x²3

Simplifying the left side:

5 = x

Therefore, the length of the box is 5 centimeters.

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The probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two is 1/25 (45/8).

The probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two is 1/25. Let's explain how to get that probability.What is the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two? Express your answer as a common fraction.A three-digit number can be represented as ABC, where A, B, and C represent the hundreds, tens, and ones digits of the number, respectively. In order for the product of the two smaller digits of the number to equal the larger digit, there are two cases to consider.

A is the largest digit. Then, A must equal B * C. C is the largest digit. Then, C must equal A * B.We must compute the number of integers for which either Case 1 or Case 2 holds. In Case 1, A can be either 2, 3, 4, 5, 6, 7, 8, or 9 (it cannot be 1 since this would make B * C = 1 and B, C have to be digits between 0 and 9). Since each of B and C can be any digit from 0 through 9, there are 10 choices for each.

Thus, there are 8 * 10 * 10 = 800 such numbers. Case 2 also has 800 integers. So, in total, there are 800 + 800 = 1600 three-digit numbers that have the desired property.There are a total of 900 three-digit numbers. Therefore, the probability of choosing a three-digit number with the desired property is 1600/900 or 8/45. Therefore, the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two is 1/25 (45/8).

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Correct Options are A, C, and F : The quadratic equations are equations in the form of [tex]ax^2[/tex] + bx + c = 0, and quadratic in form equations can be written in different forms but still have a squared term.

An equation is considered to be quadratic in form if it can be written in the general form [tex]ax^2[/tex] + bx + c = 0, where a, b, and c are constants and x is the variable.

A. [tex]x^4[/tex] – [tex]6x^2[/tex] – 27 = 0 is not quadratic in form because it contains an [tex]x^4[/tex]term, which does not fit the general form.

B. [tex]3x^4 = 2x[/tex] is not quadratic in form because it does not contain a term with [tex]x^2[/tex].

C. [tex]2(x + 5)^4 + 2x^2 + 5 = 0[/tex] is quadratic in form because it can be written as [tex]2y^4 + 2x^2 + 5 = 0[/tex], where y = x + 5.

D. [tex]6(2x + 4)^2 = (2x + 4) + 2[/tex] is not quadratic in form because the left side expands to [tex]24x^2 + 96x + 96[/tex], which does not fit the general form.

E. [tex]6x^4 = -x^2 + 5[/tex] is not quadratic in form because it contains an[tex]x^4[/tex] term and a linear term [tex](-x^2)[/tex].

F. [tex]8x^4 + 2x^2 – 4x = 0[/tex] is quadratic in form because it can be factored into [tex]2x(x + 1)(4x^2 - 1) = 0[/tex], which fits the general form.

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Select all of the following equation(s) that are quadratic in form.

A. x4 – 6x2 – 27 = 0.

B. 3x4 = 2x

C. 2(x + 5)4 + 2x2 + 5 = 0

D. 6(2x + 4)2 = (2x + 4) + 2

E. 6x4 = -x2 + 5

F. 8x4 + 2x2 – 4x = 0

Answer:

The area of the dilated flower garden is:

A' = (1/2)bh = (1/2)(2)(1.5) = 1.5 sq ft

The hypothesis testing framework is a statistical approach used to make decisions based on data. In this framework, we assume that the null hypothesis is true and then assess whether the data is likely to have arisen from a population defined by the null hypothesis.

This assessment is made by comparing a sample statistic or test statistic against its distribution under the null hypothesis.If the test statistic falls in the rejection region, this provides evidence against the null hypothesis because it suggests that our sample is unlikely to have come from a population with the parameter value(s) stated in the null hypothesis. The rejection region is defined by the significance level of the test and represents the area in which we reject the null hypothesis.
When conducting a hypothesis test, there are two types of errors that can occur: type I and type II errors. A type I error occurs when we reject a true null hypothesis, while a type II error occurs when we fail to reject a false null hypothesis.
To avoid these errors, we need to choose an appropriate level of significance (alpha) for the test and ensure that our sample size is large enough to detect a difference between the null hypothesis and the true population parameter.
In conclusion, the hypothesis testing framework is a useful tool for making decisions based on data. By assuming the null hypothesis is true and comparing our sample statistic or test statistic to its distribution under the null hypothesis, we can determine whether the data supports or contradicts the null hypothesis. If the test statistic falls in the rejection region, this provides evidence against the null hypothesis and suggests that our sample is unlikely to have come from a population with the parameter value(s) stated in the null hypothesis.

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The area A of the triangle with the given vertices is 10 square units.

Triangles have an area of A = 1/2 (b h) square units, where b and h are the triangle's base and height, respectively.

To find the area A of the triangle with the given vertices, we can use the formula:

A = 1/2 |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle. Using this formula, we get:

A = 1/2 |(02 + 55 + 10) - (05 + 51 + 20)|

= 1/2 |25 - 5|

= 1/2 |20|

= 10

As a result, the triangle with the specified vertices has a surface area of 10 square units.

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