Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. 4x² + 4 x=3 .

The factored form of the quartic function is:

f(x) = k(x - 0)(x - 6)

Here, we have,

A quartic function is a polynomial function of degree 4.

To find a quartic function with the given zeros, we can start by writing the equation in factored form using the zero-product property.

Since the zeros are x = 0 and x = 6, we can write the factors as (x - 0) and (x - 6).

The factored form of the quartic function is:

f(x) = k(x - 0)(x - 6)

To determine the value of the constant k, we need more information.

For example, if we know the value of f(1), we can substitute it into the equation and solve for k.

Without additional information, we cannot determine the specific quartic function.

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(a) The Nash equilibrium for the $20 ultimatum game is a 50-50 split: the proposer offers $10 and the responder accepts. (b) Experimental findings show deviations from the Nash equilibrium due to fairness considerations, strategic behavior, and social preferences. (c) Proposers deviate from Nash equilibrium due to fairness concerns and strategic considerations. (d) To test the explanation, conduct experiments manipulating fairness norms and incentives to observe proposers' behavior.

(a) The Nash equilibrium for the $20 ultimatum game, where offers are made to the nearest $0.50, is a 50-50 split, where the proposer offers $10 and the responder accepts.

(b) The experimental literature has found deviations from the Nash equilibrium in both proposers and responders. These deviations can be attributed to various factors such as fairness considerations, strategic behavior, and social preferences.

(c) The behavior of the proposer deviates from the Nash equilibrium due to factors like fairness concerns and strategic considerations. Proposers may make more generous offers to avoid rejection or to signal fairness and maintain a positive reputation.

(d) To test the explanation that proposers deviate from the Nash equilibrium due to fairness concerns, one could design an experiment where fairness is manipulated. Participants could be exposed to different fairness treatments, and their offers could be observed and compared to those in a control group. Statistical analysis can then be used to determine if there is a significant difference in offers between the fairness treatments and the control group.

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The square root of the given fraction simplifies to 2. A numerator of a 2 to the power of 1002 plus 2 to the power of 1006

The square root of the fraction (2^1002 + 2^1006) / (17 * 2^998) can be simplified as follows:

To simplify this expression, let's start by breaking down the numerator and the denominator separately.

Numerator:

We have 2^1002 + 2^1006. Notice that both terms have a common factor of 2^1002. Factoring that out, we get:

2^1002 * (1 + 2^4)

Simplifying further, we have:

2^1002 * (1 + 16) = 2^1002 * 17

Denominator:

The denominator is 17 * 2^998.

Now, let's combine the simplified numerator and denominator:

√((2^1002 + 2^1006) / (17 * 2^998))

= √((2^1002 * 17) / (17 * 2^998))

= √(2^1002 / 2^998)

= √(2^4)

= 2

Therefore, the square root of the given fraction simplifies to 2.

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The rectangle has a length of 7 feet and width of 4 feet. The area of a rectangle is 28 ft²

An equation is an expression that shows the relationship between two or more numbers and variables.

The question is not complete. Let us assume the area of a rectangle is 28 ft², and the length of the rectangle is 5 ft less than three times the width. Hence:

Length = l, width = w

Area = lw = 28    (1)

l = 3w - 5

But:

lw = 28;

Substitute l = 3w - 5

(3w - 5)(w) = 28

3w² - 5w - 28 = 0

w = 4 ft.

l = 3(4) - 5 = 7 feet

The rectangle has a length of 7 feet and width of 4 feet

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Answer:

40 + .15m = 50 + .10m

10 = .05m

m = 200 miles

It will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.

To determine the time it will take to save enough money, we can use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r,

where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of periods.

Given:

Desired future value (FV) = $5000,

Monthly payment (P) = $240,

Monthly interest rate (r) = 5% / 12 = 0.05 / 12 = 0.004167.

We need to solve for the number of periods (n) in the formula. Rearranging the formula, we have:

n = log(1 + (FV * r) / P) / log(1 + r),

where log denotes the logarithm with base 10.

Plugging in the given values, we get:

n = log(1 + ($5000 * 0.004167) / $240) / log(1 + 0.004167).

Using a calculator, we find that n is approximately 20.998, which rounds up to 21.

Therefore, it will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.

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The equation of the circle expressed in the form x²+y²+ax+by+c=0 is (x+3)² + (y+4)² - 100 = 0.

Center of the circle = (-3,-4)Point on the circumference of the circle = P(5,2) We know that the equation of the circle is given by: (x−a)²+(y−b)²=r² where the center of the circle is (a, b) and the radius is r.

Step 1: Find the radius of the circle using the distance formula Distance between the center of the circle and point

P = radius of the circle.

We get

r = √((-3-5)² + (-4-2)²)r = √64+36r = √100 = 10

Step 2:Find the equation of the circle substituting the center and the radius into the equation of the circle

(x−a)²+(y−b)²=r²(x-(-3))² + (y-(-4))² = 10²(x+3)² + (y+4)² = 100(x+3)² + (y+4)² - 100 = 0

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The simplified form of the expression √75 - 4√18 + 2√32 is 5√3 - 4√2.

To simplify the expression √75 - 4√18 + 2√32, we need to simplify each individual square root and then combine like terms.

Let's start by simplifying each square root term:

1. √75:

We can simplify √75 by breaking it down into its prime factors. Since 75 is divisible by 25, we have:

√75 = √(25 × 3)

    = √25 × √3

    = 5√3

2. -4√18:

Similarly, we can simplify √18:

√18 = √(9 × 2)

    = √9 × √2

    = 3√2

Therefore, -4√18 becomes -4(3√2) = -12√2

3. 2√32:

We can simplify √32:

√32 = √(16 × 2)

    = √16 × √2

    = 4√2

Now, we can rewrite the expression with the simplified square root terms:

√75 - 4√18 + 2√32

= 5√3 - 12√2 + 2(4√2)

= 5√3 - 12√2 + 8√2

Next, we combine like terms:

-12√2 + 8√2 = -4√2

Finally, the simplified expression becomes:

5√3 - 4√2

In summary, the simplified form of the expression √75 - 4√18 + 2√32 is 5√3 - 4√2.

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Planet 3 has the most circular orbit among the three, with an eccentricity of 0.2.

To figure out which circle is generally round among the three planets, we can think about their erraticism. The unconventionality of oval estimates how extended or round it is. A completely round circle has an unconventionality of 0, while higher whimsy values demonstrate more lengthened circles.

The recipe to ascertain capriciousness (e) is given by:

e = c/a

Where 'c' addresses the distance between the middle and one of the foci, and 'a' addresses the length of the semi-significant hub.

Since the sun is at one concentration and the other spotlight is on the positive x-pivot, we can expect that the distance between the middle and the other center is equivalent to the semi-significant hub (a).

Presently we should analyze the unconventionality of the three circles:

Planet 1:

Semi-significant hub (a) = 200 million kilometers

Distance to center (c) = 200 million kilometers

eccentricity (e) = c/a = 200/200 = 1

Planet 2:

Semi-significant pivot (a) = 150 million kilometers

Distance to center (c) = 100 million kilometers

Whimsy (e) = c/a = 100/150 ≈ 0.67

Planet 3:

Semi-significant pivot (a) = 250 million kilometers

Distance to center (c) = 50 million kilometers

eccentricity (e) = c/a = 50/250 = 0.2

Contrasting the eccentricities, we can see that Planet 3 has the least unpredictability of 0.2, demonstrating the most roundabout circle among the three planets.

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F(-2) simplifies to 2 to evaluate the function f(t) = t/(t + 1) at the value t = -2, we substitute t = -2 into the function and perform the calculation:

f(-2) = (-2)/((-2) + 1)

simplifying the denominator:

f(-2) = (-2)/(-1)

when dividing by a negative number, we can simplify by multiplying both the numerator and denominator by -1:

f(-2) = -2/1 = -2

so, f(-2) equals -2.

f(-2) = -2/(-1)

to evaluate the function f(t) = t/(t + 1) at the value t = -2, we substitute t = -2 into the function:

f(-2) = (-2)/((-2) + 1) = (-2)/(-1) = 2 the function f(t) = t/(t + 1) represents a rational function that is undefined when the denominator (t + 1) is equal to zero. in this case, when t = -2, the denominator is not zero, and the function evaluates to -2.

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1) The equation P = 2Q + 3 represents a supply curve. The slope of the curve is 2.

2) The equation P = -8Q + 10 represents a demand curve. The slope of the curve is -8.

3) Solving the system of equations P = 12 - 2Q and P = 3 + Q, we find that the equilibrium price is $7 and the equilibrium quantity is 5. The supply and demand curves can be graphed using the "Supply" and "Demand" tools.

4) Using the equations P = 50 - 2Qd and P = 10 + 2Qs, we determine that the equilibrium quantity is 20 units and the equilibrium price is $30. If the price changes to $20, the new quantity supplied is 10 units and the new quantity demanded is 30 units, resulting in a surplus of 20 units. To return to equilibrium, the market will experience downward pressure on price, which will decrease quantity supplied and increase quantity demanded until a new equilibrium is reached.

1) The equation P = 2Q + 3 represents a supply curve because it shows the relationship between price (P) and quantity supplied (Q). The slope of the curve is 2, indicating that for every unit increase in quantity supplied, the price increases by 2 units.

2) The equation P = -8Q + 10 represents a demand curve because it shows the relationship between price (P) and quantity demanded (Q). The slope of the curve is -8, indicating that for every unit increase in quantity demanded, the price decreases by 8 units.

3) Solving the system of equations P = 12 - 2Q and P = 3 + Q, we set them equal to each other to find the equilibrium point. By substituting P, we get 12 - 2Q = 3 + Q. Simplifying the equation, we find Q = 5. Substituting this value into either equation, we find P = 7. Therefore, the equilibrium price is $7 and the equilibrium quantity is 5. The supply and demand curves can be graphed using the given equations, and the equilibrium point can be marked as (5, 7).

4) Using the equilibrium condition Qs = Qd, we set 10 + 2Qs = 50 - 2Qd. Solving this equation, we find Q = 20. Therefore, the equilibrium quantity is 20 units. Substituting this value into either equation, we find P = $30. If the price changes to $20, we can substitute this value into the demand equation to find Qd = 30 units. Substituting into the supply equation, we find Qs = 10 units. This creates a surplus of 20 units. To return to equilibrium, the market will experience downward pressure on price as suppliers try to sell their excess supply. This will decrease quantity supplied and increase quantity demanded until a new equilibrium is reached.

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b. There are 5,040 possible combinations if Miranda can use each number only once in the four-number combination.

If Miranda can use each number only once in the four-number combination, the number of combinations possible can be calculated using the concept of permutations.

There are 10 choices for the first number, 9 choices for the second number (as one number has already been used), 8 choices for the third number (as two numbers have already been used), and 7 choices for the fourth number (as three numbers have already been used).

Therefore, the total number of combinations is calculated as:

10 × 9 × 8 × 7

= 5,040

So, there are 5,040 possible combinations if Miranda can use each number only once in the four-number combination.

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The solution to the system of equations is:

a = 2

b = -1

c = -3

To solve the system of equations using an inverse matrix, we can represent the system in matrix form:

[A] [X] = [B]

where:

[A] = coefficient matrix

[X] = variable matrix

[B] = constant matrix

The coefficient matrix [A] is:

| 1 2 1 |

| 0 1 1 |

|-3 0 1 |

The variable matrix [X] is:

| a |

| b |

| c |

The constant matrix [B] is:

| 14 |

| 1 |

| 6 |

To find [X], we need to calculate the inverse of [A] and multiply it by [B]:

[X] = [A]⁻¹ [B]

First, we find the inverse of [A]. If the inverse exists, the product [A]⁻¹ [A] should be the identity matrix [I]:

[A]⁻¹ [A] = [I]

Next, we can find the inverse of [A]:

| -1/3 2/3 -1/3 |

| 1/3 -1/3 2/3 |

| 1/3 -1/3 -1/3 |

Now, we can multiply [A]⁻¹ by [B]:

[X] = [A]⁻¹ [B]

| a | | -1/3 2/3 -1/3 | | 14 |

| b | = | 1/3 -1/3 2/3 | * | 1 |

| c | | 1/3 -1/3 -1/3 | | 6 |

Multiplying the matrices, we get:

| a | | 2 |

| b | = |-1 |

| c | |-3 |

Therefore, the solution to the system of equations is:

a = 2

b = -1

c = -3

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The expression [tex]\sqrt{3}[/tex]/ -27 no longer has an actual-quantity root due to the fact the square root characteristic and most other root functions are defined most effectively for non-negative real numbers.

To recognize the actual-wide variety root of this expression, we need to evaluate it.

The square root of 3 ([tex]\sqrt{3}[/tex]) is an irrational number, about equal to 1.732. When we divide this price by -27, the result is about -0.064.

In terms of actual-number roots, there may be no rectangular root or any other root of a negative quantity. The square root function and most different root functions are described handiest for non-bad real numbers. Therefore, in this situation, the expression [tex]\sqrt{3}[/tex]/ -27 does now not have a real-variety root.

It's worth noting that there are complex number roots for bad numbers. Complex numbers enlarge the real wide variety of gadgets by means of introducing the imaginary unit (i), allowing for the square root of negative numbers. However, within the context of this question, which specially asks for the real-range root, the answer is that there's no actual-variety root for the expression [tex]\sqrt{3}[/tex]/ -27.

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The linear factor that represents the 13-ft length is x - 5. The dimensions of the tank are 13 ft x 4 ft x 3 ft.

The polynomial 2x³ + 9x² + 4x - 15 represents the volume of the rectangular holding tank, where the depth is (x - 1) ft and the length is 13 ft. If we assume that the length is the greatest dimension, then the volume of the tank can be expressed as follows:

(length)(width)(depth) = 13x²(x - 1) = 13x³ - 13x²

Comparing this to the given polynomial, we can see that the linear factor that represents the 13-ft length is x - 5.

The dimensions of the tank can then be found by solving the equation 13x² - 13x = 2x³ + 9x² + 4x - 15. This equation can be solved as follows:

2x³ - 13x² - 13x + 15 = 0

x(2x² - 13x - 15) = 0

(x - 5)(2x + 3) = 0

x = 5 or x = -\frac{3}{2}

Since the length is the greatest dimension, the value of x must be 5. Therefore, the dimensions of the tank are 13 ft x 4 ft x 3 ft.

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The difference between the two localities is 1 speaker.

Let's solve the problems step by step:

a) To determine which of the two locations has a higher number of speakers of a language other than Spanish, we need to compare the ratios of speakers in El Cerrito and El Paseo.

In El Cerrito, 3 out of every 4 people speak a different language than Spanish. This can be written as a ratio: 3:4.

In El Paseo, 5 out of every 7 people speak a different language than Spanish. This can be written as a ratio: 5:7.

To compare the two ratios, we can find a common denominator. In this case, the least common multiple of 4 and 7 is 28.

In El Cerrito, the ratio becomes 21:28 (multiplying both sides by 7).

In El Paseo, the ratio becomes 20:28 (multiplying both sides by 4).

From the ratios, we can see that in El Cerrito, there are 21 out of 28 people who speak a different language than Spanish, while in El Paseo, there are 20 out of 28 people who speak a different language than Spanish.

Since 21 is greater than 20, El Cerrito has a higher number of speakers of a language other than Spanish.

b) The difference between the two localities can be calculated by subtracting the number of speakers in El Paseo from the number of speakers in El Cerrito.

In El Cerrito, there are 21 speakers of a different language than Spanish.

In El Paseo, there are 20 speakers of a different language than Spanish.

The difference is obtained by subtracting 20 from 21, resulting in a difference of 1 speaker.

Therefore, the difference between the two localities is 1 speaker.

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Jacques should have spent 1 hour working on problems and 3.75 hours reading to get the best possible exam score.

Let's denote the increase in exam score by x, and assume that working on 15 problems raises the exam score by x and reading the textbook for 1 hour also raises the exam score by x. We can then set up the following equation:

4 = a(15) + b

where a is the number of hours spent on problem sets, b is the number of hours spent reading the textbook, and 4 represents the total number of study hours available.

To solve for a and b, we can use the information given in the problem: "working on 15 problems raises a student’s exam score by about the same amount as reading the textbook for 1 hour." This means that:

15a = b

Substituting this into our first equation, we get:

4 = a(15) + 15a

Simplifying this equation, we get:

4 = 16a

a = 0.25

So Jacques should spend 0.25 * 4 = 1 hour working on problem sets, and 15 * 0.25 = 3.75 hours reading the textbook.

Therefore, Jacques should have spent 1 hour working on problems and 3.75 hours reading to get the best possible exam score.

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The area of the surface S, where S is the part of the plane 2x + y + 2z = 10 that lies inside the cylinder x^2 + y^2 = 16, needs to be calculated.

the area of the surface S, we first need to determine the region of the plane 2x + y + 2z = 10 that lies inside the cylinder x^2 + y^2 = 16. The given equation of the plane represents a flat surface in three-dimensional space, while the equation of the cylinder represents a circular tube.

By substituting the equations, we can find the points of intersection between the plane and the cylinder. Once we have the points of intersection, we can calculate the area of the surface S within that region. This can be done by integrating over the region or by approximating the area using numerical methods.

The calculation of the area involves working with three-dimensional geometry and finding the intersection between a plane and a cylinder. Depending on the specific values of x, y, and z in the region, the calculation may involve solving equations, using integration techniques, or applying numerical methods.

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The correct answer is B. The drop of 15 percentage points on an economics test exceeded the marginal benefit from a second hour of gymnastics.

This conclusion can be drawn by comparing the marginal benefit and marginal cost of Kate's decision to continue gymnastics for an additional hour. Marginal benefit refers to the additional benefit or satisfaction gained from an extra unit of an activity, while marginal cost refers to the additional cost or sacrifice incurred from that same unit. In this case, Kate's marginal benefit from the first hour of gymnastics was high enough to justify the time and effort invested. However, when she decided to extend her workout for a second hour and cut her study time, the marginal cost of sacrificing study time resulted in a significant drop in her economics grade. This drop of 15 percentage points on the economics test outweighed the marginal benefit she expected to gain from the extra hour of gymnastics.

The decision to continue gymnastics for the first hour had a positive marginal benefit that exceeded its marginal cost. Kate likely felt the physical and mental benefits of exercising, which enhanced her overall well-being and potentially improved her focus and energy levels for studying. However, the decision to extend the workout for a second hour resulted in a significant drop in her economics grade, indicating that the marginal cost of sacrificing study time was higher than the marginal benefit gained from the additional hour of gymnastics. This suggests that Kate should have allocated more time to studying rather than spending an extra hour on gymnastics, considering the impact it had on her academic performance.

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The rational expression (x² + 10x + 25) / (x² + 9x + 20) simplifies to (x + 5) / (x + 4), with the restriction that x cannot be equal to -4.

To simplify the rational expression (x² + 10x + 25) / (x² + 9x + 20), we can factorize the numerator and denominator and then cancel out any common factors.

Factorizing the numerator:

x² + 10x + 25 = (x + 5)(x + 5) = (x + 5)²

Factorizing the denominator:

x² + 9x + 20 = (x + 4)(x + 5)

Now, we can simplify the expression:

(x² + 10x + 25) / (x² + 9x + 20) = (x + 5)² / (x + 4)(x + 5)

After canceling out the common factor (x + 5) from the numerator and denominator, we are left with:

= (x + 5) / (x + 4)

Therefore, the simplified rational expression is (x + 5) / (x + 4). However, we need to note the restriction on the variable. In this case, x cannot be equal to -4 since it would result in division by zero, which is undefined. So, the restriction is x ≠ -4.

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The possible values of 'a' such that √72 + √a simplifies to a single term are when 'a' is equal to 2.

To simplify the expression √72 + √a into a single term, we need to find values of 'a' that allow us to combine the square roots.

First, let's simplify √72:

√72 = √(36 * 2) = √36 * √2 = 6√2

Now, our expression becomes:

6√2 + √a

To combine the square roots into a single term, the radicands (numbers inside the square root) must be the same. In this case, the radicands are 2 and 'a'. To simplify the expression, we need 'a' to be equal to 2.

Therefore, the possible values of 'a' such that √72 + √a simplifies to a single term are when 'a' is equal to 2.

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Since the value of x is not provided in the question, we cannot determine the exact measure of angle 9 without more information. On the other hand, the measure of angle 10 is given as 2. Therefore, we know that m∠ 10 = 2.

The measure of angle 9 is given by the expression 2x - 4, where x represents an unknown value. The measure of angle 10 is stated as 2, without any variable involved. To find the measure of angle 9, we need to substitute the given value of x into the expression 2x - 4 and simplify the equation. The answer will be a numerical value representing the measure of angle 9.

In summary, we need to substitute the value of x into the expression 2x - 4 to determine the measure of angle 9. The measure of angle 10 is already given as 2.

Given that m∠ 9 = 2x - 4, we need to substitute the value of x into this expression to find the measure of angle 9. Since the value of x is not provided in the question, we cannot determine the exact measure of angle 9 without more information.

On the other hand, the measure of angle 10 is given as 2. Therefore, we know that m∠ 10 = 2.

To solve for the measure of angle 9, we would need the value of x. Without it, we cannot calculate the precise measure of angle 9. However, if the value of x were given, we could substitute it into the expression 2x - 4 to find the numerical value representing the measure of angle 9.

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The required equation of the circle is x² + y² - 18x + 56 = 0.

The equation of a circle is calculated using the formula -

(x - h)² + (y - k)² = r²

Keep the values in formula where h and k are the coordinates of the centre of the circle. Moreover, r represents the radius of the circle.

(x - 9)² + (y - 0)² = 5²

(x - 9)² + y² = 25

Expanding the x-axis coordinates of the equation

x² + 81 - 18x + y² = 25

x² + y² - 18x + 81 - 25 = 0

Subtracting the relevant values in the equation

x² + y² - 18x + 56 = 0

Hence, the required equation of the circle is x² + y² - 18x + 56 = 0.

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Given that the common chord AB between circles P and Q is perpendicular to the segment connecting the centers of the circles, we can form a right triangle with AB as the hypotenuse and the segment connecting the centers as one of the legs. Let O₁ and O₂ be the centers of circles P and Q, respectively.

Since AB is the common chord, it is also the diameter of both circles. Let r₁ and r₂ be the radii of circles P and Q, respectively. Therefore, AB = 2r₁ = 2r₂ = 10. Let d be the distance between the centers O₁ and O₂. The segment connecting the centers is the other leg of the right triangle. Since AB is perpendicular to this segment, the right triangle formed is a right triangle with sides AB, r₁, and d.

Using the Pythagorean theorem, we can write the equation:

(2r₁)² = r₁² + d²

Simplifying, we have:

4r₁² = r₁² + d²

3r₁² = d²

Substituting AB = 10 and r₁ = r₂ = 5, we get:

3(5)² = d²

75 = d²

Taking the square root of both sides, we find:

d = √75 = √(25 * 3) = 5√3 Therefore, the length of PQ is equal to the distance between the centers of the circles, which is 5√3.

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The table of second differences for each polynomial function is given below and  the second differences of any quadratic function will be a constant value is the conjecture.

To find the second differences for the polynomial function y = 5x², we need to calculate the first differences and then the differences of those first differences.

x            y = 5x² First Differences Second Differences

0        0                  -                          -

1         5                  5                         -

2       20                 15                       10

3       45                 25                       10

4       80                 35                       10

The first differences are obtained by subtracting the previous value of y from the current value of y.

The second differences are then calculated by finding the differences between consecutive first differences.

1st Second Difference: 10

2nd Second Difference: 10

3rd Second Difference: 10

From the table, we can observe that the second differences for the quadratic function y = 5x² are all the same.

Based on this observation, we can make a conjecture that the second differences of any quadratic function will be a constant value.

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The r-value that represents the most moderate correlation is given as follows:

0.18.

A correlation coefficient is a statistical measure that indicates the strength and direction of a linear relationship between two variables.

The coefficients can range from -1 to +1, with -1 indicates a perfect negative correlation, 0 indicates no correlation, and +1 indicates a perfect positive correlation.

Hence the most moderate correlation is the value with the absolute value closest to zero, hence it is given as follows:

0.18.

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The speed of the train is 45 mi/h hence the correct answer is option G. 45 mi/h.

Let's assume the speed of the train is x mi/h.

Since the car is traveling 15 mi/h faster than the train, the speed of the car is (x + 15) mi/h.

The distance traveled by train in 2 hours is 2x miles (since speed * time = distance).

The distance traveled by the car in 2 hours is 2(x + 15) miles.

According to the Pythagorean theorem, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, the hypotenuse represents the distance between the train and the car, which is approximately 150 miles.

Using the Pythagorean theorem, we can set up the equation:

[tex](2x)^2 + [2(x + 15)]^2 = 150^2\\4x^2 + 4(x + 15)^2 = 150^2[/tex]

Simplifying and solving the equation, we find:

[tex]4x^2 + 4(x^2 + 30x + 225) = 22500\\4x^2 + 4x^2 + 120x + 900 = 22500\\8x^2 + 120x + 900 - 22500 = 0\\8x^2 + 120x - 21600 = 0[/tex]

Dividing the equation by 8 to simplify:

[tex]x^2 + 15x - 2700 = 0[/tex]

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Factoring is the easiest method in this case:

(x + 60)(x - 45) = 0

Setting each factor equal to zero, we have:

x + 60 = 0  or  x - 45 = 0

x = -60  or  x = 45

Since the speed cannot be negative, we discard the solution x = -60.

Therefore, the speed of the train is 45 mi/h hence the correct answer is option G. 45 mi/h.

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Answer:

  see attached

Step-by-step explanation:

You want the steps to simplify the sum of two rational expressions.

In general, the steps will be ...

See the attachment for the order for the given sum.

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Additional comment

You can also look for factors in the combined numerator that will cancel denominator factors. And, you can expand the denominator to standard form.

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