suppose that X is uniformly distributed on the finite set {6,7,8,9}. Suppose Y is uniformly distributed on the finite set {18,…,26}. Suppose X and Y are independent.
(a) The moment generating function of X is Mx(t)=
(b) The moment generating function of X+Y is MX+Y(t)=

The volume of the solid obtained by rotating the region under the graph of f(x) = 9[tex]x^3[/tex] about the y-axis over the interval [0, 1] is 18π/5 cubic units.

To find the volume of the solid obtained by rotating the region under the graph of f(x) = 9[tex]x^3[/tex] about the y-axis over the interval [0, 1], we can use the method of cylindrical shells.

The volume of a solid obtained by rotating a region about the y-axis can be calculated by integrating the circumference of each cylindrical shell multiplied by its height.

In this case, the height of each cylindrical shell is given by the function f(x) = 9[tex]x^3[/tex], and the radius is the x-value at each point along the interval [0, 1].

Using the formula for the volume of a cylindrical shell, the volume V can be expressed as:

V = ∫[0, 1] 2πx(9[tex]x^3[/tex]) dx

Simplifying the equation, we have:

V = 18π ∫[0, 1] [tex]x^4[/tex] dx

Integrating [tex]x^4[/tex] with respect to x, we get:

V = 18π × ([tex]x^5[/tex]/5) | [0, 1]

Plugging in the limits of integration, we have:

V = 18π × (([tex]1^5[/tex]/5) - ([tex]0^5[/tex]/5))

V = 18π × (1/5)

V = 18π/5

Therefore, the volume of the solid obtained by rotating the region under the graph of f(x) = 9[tex]x^3[/tex] about the y-axis over the interval [0, 1] is 18π/5 cubic units.

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Answer:

-2/3

Step-by-step explanation:

To determine the gradient of the line PS, we can rearrange the equation 3y + 2x = 6 into the slope-intercept form, y = mx + c, where m is the gradient.

Rearranging the equation:

3y + 2x = 6

3y = -2x + 6

y = (-2/3)x + 2

Comparing this equation with y = mx + c, we can see that the gradient (m) of line PS is -2/3.

Therefore, the gradient of line PS is -2/3.

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The option that proves that The set of integers is open under division is 46/16. Option B

The set of integers is not closed under division. Unlike under addition, subtraction and multiplication.

This means that dividing two integers does not always result in an integer.

"open" would most likely mean mean the same as "not closed."

The example that proves this statement is 46/16, as this does not result in an integer. An integer is a whole number. When you divide 46 by 16, it does not give a whole number.

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A sphere thrown upward at a 45-degree angle to the horizontal in a classroom elastically bounces off the ceiling while still rising

At time t = 0, a sphere is launched with an initial velocity at a 45-degree angle to the horizontal in a classroom. The sphere follows a parabolic trajectory as it rises due to the upward component of its initial velocity and experiences the downward pull of gravity. While the sphere is still ascending, it reaches the ceiling and collides with it.

During the elastic collision, the sphere's motion is reversed. It rebounds off the ceiling, changing its direction but maintaining its kinetic energy. As a result, the sphere starts descending with the same speed it had before the collision but in the opposite direction. The angle of descent will also be 45 degrees to the horizontal, mirroring the angle of the initial launch.

Throughout the entire process, neglecting air resistance, the total mechanical energy of the sphere is conserved since the collision with the ceiling is elastic.

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Uncle Makhosi needs approximately 1.56 kilograms of butter.

To determine the amount of butter in kilograms, we'll use the conversion rate of 1 kg = 2.25 pounds.

First, we need to convert 3 and a half pounds to a decimal form. Since half a pound is equal to 0.5 pounds, we can express 3 and a half pounds as 3.5 pounds.

Next, we'll use the conversion rate to calculate the equivalent weight of 3.5 pounds in kilograms:

3.5 pounds * (1 kg / 2.25 pounds) = 1.56 kilograms (rounded to two decimal places).

To summarize, based on the given conversion rate of 1 kg = 2.25 pounds, Uncle Makhosi requires approximately 1.56 kilograms of butter to fulfill his 3 and a half pound requirement. This conversion can be useful when dealing with different units of measurement, allowing us to easily switch between kilograms and pounds.

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a) To determine the resultant R of the three forces F, P, and Q, we perform vector addition separately. The x-component of the resultant is the sum of the x-components of the individual forces, and the y-component of the resultant is the sum of the y-components of the individual forces.

b) To minimize the magnitude of the resultant by adding an additional force T in the ±X direction, we need to set the x-component of the resultant to zero. This means the sum of the x-components of the forces, including T, should equal zero. Solving this equation will give us the magnitude T that minimizes the resultant.

a) To find the resultant R, we add the x-components and y-components separately. The x-component of the resultant is the sum of the x-components of the forces (Fx + Px + Qx), and the y-component of the resultant is the sum of the y-components of the forces (Fy + Py + Qy).

b) To minimize the magnitude of the resultant, we set the x-component of the resultant to zero. This means that the sum of the x-components of the forces, including the additional force T in the ±X direction, should equal zero (Fx + Px + Qx + Tx = 0). By solving this equation, we can determine the magnitude T that minimizes the resultant.

Please note that without specific values for F, P, Q, and the angles involved, we cannot provide a numerical answer. The solution will require substituting the given values into the equation and solving for T.

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Mathematics is a fundamental subject in education that is essential in the development of an individual’s critical thinking and problem-solving skills. It is one of the primary subjects that are taught in elementary schools and must be taught appropriately for children to understand and gain the required skills. One true statement about mathematics teaching in elementary schools is that it should be student-centered.

Teachers should focus on the child's understanding, individual learning abilities and styles, and progress.The learning experience should not be centered on the teacher but on the students. Teachers should also make the subject engaging and interactive to enable students to grasp the content easily and maintain their interest in the subject. Besides, teachers should develop a personalized learning approach that allows students to learn and understand the subject at their own pace while creating opportunities for individual and group learning. Through such learning approaches, students can develop the required mathematical concepts and competencies such as critical thinking, problem-solving, and decision-making skills that will be beneficial in their future endeavors.

In elementary schools, mathematics teaching should be student-centered. It should be engaging and interactive to enable students to grasp the content easily and maintain their interest in the subject.

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Yes, I can find the slope of a line and type the correct code. The SLOPE function will return the slope of the linear regression line that best fits the data.

The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The formula for finding the slope of a line is (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line.

The slope is a measure of how steep the line is. It can be positive, negative, zero, or undefined. The code for finding the slope of a line in ALL CAPS with no spaces is SLOPE.

To use this function in Excel, you need to provide the range of x-values and the range of y-values.

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In general, approximately 78% of the values in a data set lie at or below the 78th percentile. Approximately 48% of the values in a data set lie at or above the 52nd percentile. If a sample consists of 600 test scores, approximately 68% of them would be at or below the 68th percentile. If a sample consists of 600 test scores, approximately 6% of them would be at or above the 6th percentile.

Percentiles are used to divide a dataset into equal parts based on the values. The nth percentile represents the value below which n% of the data falls.

In the first statement, we can say that approximately 78% of the values in a data set lie at or below the 78th percentile. This means that if we arrange the data in ascending order, the 78th percentile will be the value below which 78% of the data falls.

In the second statement, we can say that approximately 48% of the values in a data set lie at or above the 52nd percentile. This means that if we arrange the data in ascending order, the 52nd percentile will be the value above which 48% of the data falls.

For the third and fourth statements, we can apply the same concept. If a sample consists of 600 test scores, approximately 68% of them would be at or below the 68th percentile. This means that if we arrange the 600 test scores in ascending order, the 68th percentile will be the score below which 68% of the scores fall.

Similarly, if a sample consists of 600 test scores, approximately 6% of them would be at or above the 6th percentile. This means that if we arrange the 600 test scores in ascending order, the 6th percentile will be the score above which 6% of the scores fall.

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The student's grade on the examination was transformed to a z-value of 0.67, indicating that she scored approximately in the top one-third of the distribution.

In a normal distribution, z-scores represent the number of standard deviations a particular value is from the mean. A z-score of 0.67 corresponds to a location that is about two-thirds of a standard deviation above the mean. Since the normal distribution is symmetric, we can infer that the student's score is higher than about two-thirds of the scores in the distribution.

To understand this further, let's consider the properties of the normal distribution. In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, the area under the curve between the mean and a z-score of 0.67 is approximately one-third. This means that the student's score falls within the top one-third of the distribution. However, it's important to note that without knowing the exact details of the distribution of scores and its mean and standard deviation, we cannot provide precise information about the student's ranking among her peers. Nonetheless, based on the given z-value, we can conclude that she performed quite well on the examination.

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True. Javier takes a shower to save time. He gets into the shower at 6:50 and is out by 7:10. When he used to take baths, it would take him a quarter of an hour.

The statement is true. Javier takes a shower to save time. When he gets into the shower at 6:50, he is out by 7:10. This implies that he spends 20 minutes in the shower.

On the other hand, when he used to take baths, it would take him a quarter of an hour. A quarter of an hour is equivalent to 15 minutes. Therefore, Javier's bath time used to be 15 minutes.

Comparing the time spent in the shower (20 minutes) with the time spent in the bath (15 minutes), we can see that taking a shower saves Javier 5 minutes of time compared to taking a bath.

Overall, the statement confirms that Javier chooses to take a shower to save time. By opting for a shower instead of a bath, he reduces the amount of time spent on personal hygiene, saving approximately 5 minutes in this scenario.

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The given statement " When the film is placed into the XCP (extension cone paralleling) holder with the smooth side of the film towards the throat, after processing, it will appear darker" is false because it will  lighter, not darker.

The smooth side of the film is the side that interacts with the X-ray radiation and receives the image, while the emulsion side contains the light-sensitive crystals that react to the radiation.

Placing the smooth side towards the throat ensures that the image is sharp and clear, as the X-ray beam travels through the teeth and soft tissues before reaching the film.

After processing, the exposed areas of the film turn dark, representing the captured X-ray image, while the unexposed areas remain light or clear.

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The greater value of the x-intercept of g(2) rounded to the nearest hundredth is 14.89.

What is a definite integral?

To find the definite integral using its geometric interpretation, we need to sketch the graph of the integrand and calculate the area under the curve between the given limits.

First, let's sketch the graph of the integrand function, g(x). The integrand is given as g(x) = 160 - 18|x - 6|. We'll break it down into two cases based on the absolute value:

For x ≤ 6:

In this case, |x - 6| = -(x - 6) = -x + 6. Therefore, g(x) = 160 - 18(-x + 6) = 160 + 18x - 108 = 18x + 52.

For x > 6:

In this case, |x - 6| = x - 6. Therefore, g(x) = 160 - 18(x - 6) = 160 - 18x + 108 = -18x + 268.

Now let's plot the graph of g(x):

The graph of g(x) consists of two line segments: one with a positive slope of 18 starting at (0, 52) and passing through (6, 160), and the other with a negative slope of -18 starting at (6, 160) and passing through (14.89, 0).

To find the definite integral of g(x) dx between x = -6 and x = 2, we need to calculate the area under the curve bounded by these limits. Since the area below the x-axis represents negative values, we'll subtract the area of the triangle below the x-axis from the area of the trapezoid above the x-axis.

The area of the trapezoid is given by:

A = [(b1 + b2) * h] / 2,

where b1 is the length of the longer base, b2 is the length of the shorter base, and h is the height.

In this case, the longer base is the segment from (6, 160) to (14.89, 0), which has a length of 8.89 units. The shorter base is the segment from (0, 52) to (6, 160), which has a length of 6 units. The height is the difference between the y-coordinates of the two bases, which is 160 - 0 = 160 units.

Therefore, the area of the trapezoid is:

A = [(8.89 + 6) * 160] / 2 = 1758.4 square units.

Now let's calculate the area of the triangle below the x-axis. The base of the triangle is 6 units, and the height is 52 units.

Therefore, the area of the triangle is:

A = (6 * 52) / 2 = 156 square units.

To find the definite integral, we subtract the area of the triangle from the area of the trapezoid:

Definite integral = Area of trapezoid - Area of triangle

= 1758.4 - 156

= 1602.4 square units.

Since the question asks for the greater value of the x-intercept of g(2), let's find the x-coordinate of the point

We know that g(x) = 18x + 52 for x ≤ 6, and g(x) = -18x + 268 for x > 6.

For x ≤ 6:

Setting g(x) = 0, we have:

18x + 52 = 0

18x = -52

x = -52/18 ≈ -2.89

For x > 6:

Setting g(x) = 0, we have:

-18x + 268 = 0

18x = 268

x = 268/18 ≈ 14.89

Since we're interested in the greater value of the x-intercept, the x-coordinate of the point where g(2) intersects the x-axis is approximately 14.89 (rounded to the nearest hundredth).

Therefore, the greater value of the x-intercept of g(2) rounded to the nearest hundredth is 14.89.

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To match the expression with the correct equivalent expression, let's simplify each expression:

Expression 1: 2^4

Expression 2: (2^2) * (2^-2)

Now, let's simplify each expression:

Expression 1: 2^4 = 2 * 2 * 2 * 2 = 16

Expression 2: (2^2) * (2^-2) = (2 * 2) * (1 / (2 * 2)) = 4 * (1 / 4) = 1

Matching the expressions with their equivalent values:

Expression 1: 2^4 --> 16

Expression 2: (2^2) * (2^-2) --> 1

Therefore, the correct matches are:

2^4 --> 16

(2^2) * (2^-2) --> 1

The value of the horizontal component is 6.75 Newtons.

Let's assume the horizontal component of the force is represented by 'x' (in Newtons).

According to the given information, the vertical component is four-thirds (4/3) of the horizontal component. Mathematically, this can be expressed as:

Vertical component = (4/3) * Horizontal component

Given that the vertical component is 9N, we can substitute it into the equation:

9N = (4/3) * x

To find the value of the horizontal component, we can rearrange the equation and solve for 'x':

x = (3/4) * 9N

x = (27/4) N

As a result, the horizontal component has a value of 6.75 Newtons.

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The correct answer is: C) virtual and upright

To determine the character of the image formed by a concave mirror, we can use the mirror formula:

1/f = 1/v - 1/u

Where:

f is the focal length of the mirror,

v is the image distance (positive for a real image and negative for a virtual image),

u is the object distance (positive when the object is in front of the mirror and negative when it's behind the mirror).

Given:

Object distance (u) = -50 cm (since the object is located in front of the mirror)

Radius of curvature (R) = -60 cm (negative for a concave mirror)

We know that the focal length (f) for a concave mirror is half the radius of curvature, so:

f = R/2 = -60/2 = -30 cm

Substituting the values into the mirror formula, we get:

1/-30 = 1/v - 1/-50

Simplifying the equation gives:

-1/30 = 1/v + 1/50

To solve for v, we can find the least common denominator and multiply all terms by 150v:

-5v = 150 - 3v

Bringing the terms with v on one side and constants on the other side:

-5v + 3v = 150

-2v = 150

v = -150/2

v = -75 cm

Since the image distance (v) is negative, the image formed by the concave mirror is virtual.

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The standard error of the regression (se) is calculated by taking the square root of the mean squared error (MSE) divided by the degrees of freedom (df).

The mean squared error (MSE) is calculated by summing the squared residuals (the differences between the actual observed values and the predicted values) and dividing by the number of observations minus the number of predictors (variables) in the regression model.

The formula for the mean squared error is:

MSE = Σ(residuals^2) / (n - k)

where Σ denotes the sum, residuals^2 represents the squared residuals, n is the number of observations, and k is the number of predictors in the regression model.

To calculate the standard error of the regression, se, we take the square root of the mean squared error divided by the degrees of freedom:

se = √(MSE / df)

The degrees of freedom (df) in a regression model is equal to the number of observations minus the number of predictors (k).

Therefore, the standard error of the regression, se, is calculated by taking the square root of the mean squared error divided by the degrees of freedom (df).

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Answer:

D

Step-by-step explanation: Slope intercept form meaning y= mx+b. mx= the slop and b=y intercept. Simply take the two points and subtract them to get the slope. Use the slope to find the y intercept.

The Graph line either by using the table or by using the equation f(x) = -0.5x + 3.

1)f(x) =  -0.5x + 3, is the equation of the form y = mx + b

2) y = mx + b is slope-intercept  equation of a line where the slope is m and the y-intercept is b, so, f(x) = - 0.5x + b has slope m = -0.5 and y-intercept b = 3.

3) To graph f(x) = -0.5x + 3, follow these steps:

draw two perpedicular axis: vertical axis, labeled y, and horizontal axis, labeled x.

draw marks on each axis, each mark equivalent to one unit.

the intersection point of the vertical and horizontal axis is the origin, i.e. point (0,0).

you can make a table with two or more points:

       x       f(x) = - 0.5x + 3

       -2         4

       0          3

       2          2

       4           1

       6          0

4) You can see the graph in the figure attached, and select any of the points on the line either by using the table or by using the equation f(x) = -0.5x + 3.

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Answer: To model the problem, we can use the equation:

p = 8 * 4

This equation represents the total number of bottles of paint, p, as the product of the number of caddies, which is 8, and the number of bottles of paint per caddy, which is 4.

To determine which function is most likely increasing quadratically, we can compare the differences in the values of f(x) and g(x) for increasing values of x.

Looking at the table, we can see that as x increases, the values of f(x) and g(x) also increase. However, the rate of increase for f(x) appears to be larger than the rate of increase for g(x).

For example, when x increases from 2 to 3, f(x) increases by 6 (from 9 to 15), while g(x) increases by 18 (from 9 to 27). Similarly, when x increases from 4 to 5, f(x) increases by 10 (from 23 to 33), while g(x) increases by 162 (from 81 to 243).

Based on these differences, it seems that f(x) is growing faster than g(x) as x increases. This suggests that f(x) is most likely increasing quadratically, while g(x) is increasing at a slower rate.

Therefore, the correct answer is:

c) f(x), because it grows faster than g(x)

Answer:

(x + 1)^2 + (y - 1)^2 =16.

Step-by-step explanation:

That would be

(x + 1)^2 + (y - 1)^2 = 4^2

---> (x + 1)^2 + (y - 1)^2 =16.

It follows from the triangle inequality that e) False, (f) False, (g) True.

(e) The statement is False. The equation ||u - v|| = ||u|| - ||v|| is not generally true. The correct equation is ||u - v|| = ||u|| + ||v||, which follows from the triangle inequality.

(f) The statement is False. The equation 3x = 7 in Z₁₁ does not have a unique solution. In Z₁₁, we have to find a number x such that 3x is congruent to 7 modulo 11. However, there is no integer solution for x in this case, so there is no unique solution.

(g) The statement is True. The equation 2x = 7 in Z₁₀ has a unique solution. In Z₁₀, the possible values for x are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Among these values, only x = 7 satisfies the equation 2x = 7, so there is a unique solution in Z₁₀.


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The simplified expression for dx is -2 / (2√(x-x²)) and the value of dx when x = 0.5 is approximately 0.877.

To find the simplified expression for dx, we'll differentiate the given function f(x) = sin(x). Since the derivative of sin(x) is cos(x), the expression for dx simplifies to:

dx = cos(x)

Since no specific value or condition is given for x, we can leave it as cos(x) without further simplification.

(d) To find the value of dx when x = 0.5, we substitute the value of x into the expression for dx, which is cos(x):

dx = cos(0.5)

Evaluating cos(0.5) using a calculator or trigonometric table, we find that cos(0.5) is approximately 0.877. Hence, the value of dx when x = 0.5 is approximately 0.877.

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Answer:

Step-by-step explanation:

You want the sine and tangent of the 2nd-quadrant angle whose cosine is -3/4.

The relevant trig identities are ...

  sin(θ) = ±√(1 -cos(θ)²) . . . . . the + sign applies in the 2nd quadrant

  tan(θ) = sin(θ)/cos(θ)

Using the given value of cosine, we find the sine to be ...

  sin(θ) = √(1 -(-3/4)²) = √(7/16)

  sin(θ) = (√7)/4

and

  tan(θ) = sin(θ)/cos(θ) = ((√7)/4)/(-3/4)

  tan(θ) = (-√7)/3

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Answer:9/1

Step-by-step explanation:

9

Step-by-step explanation:

How many six can you see in fifty four =9

Answer:

x is about 10.68 in

Step-by-step explanation:

The formula for area of a rectangle is given by the formula:

A = lw, where

Step 1:  We can plug in 260 for A, (2x + 3) for l, and x for w:

260 = (2x + 3)(x)

Step 2:  Multiply x and (2x + 3)

260 = x * 2x + x * 3

260 = 2x^2 + 3x

Step 3:  Subtract 260 from both sides to get a regular quadratic equation in standard form:

(260 = 2x^2 + 3x) - 260

0 = 2x^2 + 3x - 260

Currently 2x^2 + 3x - 260 is in standard form, whose general equation is given by:

0 = ax^2 + bx + c

We can solve for the roots of the quadratic equation using the quadratic formula, which is given by:

[tex]x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}[/tex], where

Since we know that 2 is our a value, 3 is our b value, and -260 is our c value, we plug these in for a, b, and c in the quadratic formula:

Positive answer:

[tex]x=\frac{-3+\sqrt{3^2-4(2)(-260)} }{2(2)}\\ x=\frac{-3+\sqrt{9+2080} }{4}\\ x=-3/4+1/4\sqrt{2089}\\ x=10.67639488\\x = 10.68[/tex]

If you were to find the negative answer, you'd get x is about -18.93.  Because we can't have a negative answer, we must use the positive answer and thus x is about 10.68

Optional Step 3:  We can check that x = 10.68 is the correct answer by plugging in 10.68 for x in the formula A = (2x + 3)(x) and check that we get 260 or something very close to it:

260 = (2 * 10.68 + 3)(10.68)

260 = (21.36 + 3)(10.68)

260 = (24.36)(10.68)

260 > 249.4848

Since we rounded x to the nearest hundredth, you get an approximate answer.  You'd get an exact answer if you used the unrounded answer, which I did on my graphing calculator.  I got 249.4848 when I used the rounded answer for x and 260 when I used the exact answer for x.

By completing the square, the roots of the quadratic equations are:

Case A: x = 2 ± i √5 / 2

Case B: x = - 1 / 3 ± i√(2 / 3)

In this problem we find two cases of quadratic equations, whose roots can be found by completing the square, that is, by modifying part of the quadratic equation into a perfect square trinomial and clear the resulting variable. This can be done by means of algebra properties:

Case A

- 2 · x² + 8 · x - 3 = 0

- 2 · (x² - 4 · x + 3 / 2) = 0

- 2 · (x² - 4 · x + 4) = 5 / 2

- 2 · (x - 2)² = 5 / 2

(x - 2)² = - 5 / 4

x - 2 = ± i √5 / 2

x = 2 ± i √5 / 2

Case B

3 · x² + 2 · x + 7 / 3 = 0

3 · [x² + (2 / 3) · x + 7 / 9] = 0

3 · [x² + (2 / 3) · x + 1 / 9] = - 3 · (6 / 9)

3 · (x + 1 / 3)² = - 2

x + 1 / 3 = ± i√(2 / 3)

x = - 1 / 3 ± i√(2 / 3)

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The following equation models a nonlinear function: y = x(1 + x) + 4.

A nonlinear function is a function whose plotted graph does not form a straight line but a curved line. The exponent of the variable in a nonlinear equation is greater than 1.

Here, y=-5x+7 is a linear function.

y= x/5 + 10 is a linear function.

y = x+ 10 is a linear function.

3y = 6x-9 is a linear function.

y= x(1 + x) + 4 is a quadratic function.

Therefore, the following equation models a nonlinear function: y = x(1 + x) + 4.

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