Find the area of the region bounded by the graphs of f(x) = x^3 + x^2 - 6x and g(x) = 2x - x^2

Answer: 74

Step-by-step explanation:

The volume of the pyramid can be found using the formula:

Volume = (1/3) x Base Area x Height

To find the base area, we need to find the area of the triangular base. The area of a triangle can be found using the formula:

Area = (1/2) x Base x Height

Substituting the given values, we have:

Area = (1/2) x 8 x 6 = 24 square inches

To find the height of the pyramid, we can use the Pythagorean theorem. The slant height and one-half of the base form a right triangle, so we have:

Height^2 = (Slant Height)^2 - (1/2 x Base)^2

Height^2 = 10^2 - 4^2

Height^2 = 84

Height = √84 ≈ 9.165 inches

Now we can substitute the values into the formula for the volume:

Volume = (1/3) x Base Area x Height

Volume = (1/3) x 24 x 9.165

Volume ≈ 73.96 cubic inches

Therefore, the volume of the pyramid is approximately 73.96 cubic inches.Step-by-step explanation:

The vertex of point A' in the dilated triangle A'B'C' is (6, 4.5).

1. Start by calculating the distance between the vertices of the original triangle ABC:

  - Distance between A(4, 3) and B(3, -2):

    Δx = 3 - 4 = -1

    Δy = -2 - 3 = -5

    Distance = √((-[tex]1)^2[/tex] + (-[tex]5)^2[/tex]) = √26

  - Distance between B(3, -2) and C(-3, 1):

    Δx = -3 - 3 = -6

    Δy = 1 - (-2) = 3

    Distance = √((-6)² + 3²) = √45 = 3√5

  - Distance between C(-3, 1) and A(4, 3):

    Δx = 4 - (-3) = 7

    Δy = 3 - 1 = 2

    Distance = √(7² + 2²) = √53

2. Apply the scale factor of 1.5 to the distances calculated above:

  - Distance between A' and B' = 1.5 * √26

  - Distance between B' and C' = 1.5 * 3√5

  - Distance between C' and A' = 1.5 * √53

3. Determine the coordinates of A' by using the distance formula and the given coordinates of A(4, 3):

  - A' is located Δx units horizontally and Δy units vertically from A.

  - Δx = 1.5 * (-1) = -1.5

  - Δy = 1.5 * (-5) = -7.5

  - Coordinates of A':

    x-coordinate: 4 + (-1.5) = 2.5

    y-coordinate: 3 + (-7.5) = -4.5

4. Thus, the vertex of point A' in the dilated triangle A'B'C' is (2.5, -4.5).

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Answer: the answer is -4

Step-by-step explanation: the reason why the answer is -4 becaue all of the option is in correct.

The number line inequality that represents x > -7 is: Option D

In number line inequalities we know that:

A closed circle indicates "greater than or equal to" or "less than or equal to" .

Meanwhile an open circle indicates "greater than" or "less than".

A closed circle pointing to the right indicates  "greater than or equal to" while a closed circle pointing to the left indicates "less than or equal to,"

Similarly:

An open circle pointing to the right indicates "greater than" while An open circle pointing to the left indicates  "less than".

Thus , the correct number line that shows x > -7 is: Option D

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Answer:

Relative maximum at x=0; Relative minimum at x=8/3

Step-by-step explanation:

To find the relative maximums and the relative minimums, you must first find the first derivative of the function. The first derivative of this function is 6x^2-16x. Simply it and you get 2x(3x-8). X would be equal to 0 and 8/3. Next, make a number line where you put 0 and 8/3 have a value of zero.

        +                           -                                 +      

-------------------0----------------------------8/3-----------------------

Plug in a value of x<0 to get the region left of 0. Say we use -1, we get -2(-3-8), which is positive, meaning that it is increasing there. From 0 to 8/3, if we use 1, we get 2(3-8), which is decreasing. If we use 3, we get 6(9-8), which is increasing. From this, we can see that when x=0, the graph has a relative maximum. When x=8/3, the graph has a relative minimum.

Answer:

Step-by-step explanation:

Let's represent the number of people that Ms. Hernandez can bring to the zoo as "x". Each person will require an admission ticket, which costs $10.40 per person. Additionally, there is a parking fee of $3.

The total cost of admission tickets and parking is given as $150. We can set up the equation as follows:

10.40x + 3 = 150

To solve for x, we need to isolate the variable:

10.40x = 150 - 3

10.40x = 147

Now, divide both sides of the equation by 10.40 to solve for x:

x = 147 / 10.40

Using a calculator, we find:

x ≈ 14.13

Since we can't have a fractional number of people, we need to round down to the nearest whole number since we can't bring a fraction of a person. Therefore, Ms. Hernandez can bring 14 people to the zoo, including herself.

The average speed of a car is calculated by dividing the total distance traveled by the total time taken. In this case, the car travels 50+ miles in "of an hour".

To calculate the average speed in miles per hour, we need to determine the value of "of an hour". If the value is given as a fraction, we need to convert it to a decimal.

Assuming "of an hour" is 1/2 (0.5), the average speed can be calculated as:

Average speed = Total distance / Total time

Average speed = 50+ miles / (1/2) hour

To divide by a fraction, we can multiply by its reciprocal:

Average speed = 50+ miles * (2/1) hour

Average speed = 100+ miles per hour

Therefore, the average speed of the car is 100+ miles per hour.

Remember to use the appropriate units for measurements when calculating surface area.

To find the surface area of an object, you need to calculate the total area of all its exposed surfaces. The method for finding the surface area will vary depending on the shape of the object. Here are some common shapes and their respective formulas:

1. Cube: The surface area of a cube can be found by multiplying the length of one side by itself and then multiplying by 6 since a cube has six equal faces. The formula is: SA = 6s^2, where s is the length of a side.

2. Rectangular Prism: A rectangular prism has six faces, each of which is a rectangle. To find the surface area, calculate the area of each face and add them up. The formula is: SA = 2lw + 2lh + 2wh, where l, w, and h represent the length, width, and height of the prism.

3. Cylinder: The surface area of a cylinder includes the area of two circular bases and the area of the curved side. The formula is: SA = 2πr^2 + 2πrh, where r is the radius and h is the height.

4. Sphere: The surface area of a sphere can be found using the formula: SA = 4πr^2, where r is the radius.

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The solution to the given equation is x = -1/4.To solve the equation, let's break down the steps as outlined by Deepak:

Combine like terms: Starting with the left side of the equation, combine the x terms and the constant terms separately. On the left side, we have -3x and 4x, which can be combined to give x. Similarly, we have -5 and 5, which cancel each other out, leaving us with zero.

Simplify both sides: Now, the equation becomes x = -7/4 - 3x.

Move all the x terms to one side: To isolate the x term on one side, we can add 3x to both sides of the equation. This gives us 4x + 3x = -7/4.

Combine like terms: On the left side, we have 4x and 3x, which can be added to give 7x. The equation now becomes 7x = -7/4.

Solve for x: To solve for x, we divide both sides of the equation by 7. This yields x = -1/4.

Therefore, the solution to the given equation is x = -1/4.

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The expression that represents the statement "Divide the difference of 27 and 3 by the difference of 16 and 4" is (27 - 3) / (16 - 4), which simplifies to 2.

The statement "Divide the difference of 27 and 3 by the difference of 16 and 4" can be represented using algebraic expressions.

To find the difference between two numbers, we subtract one from the other. So, the difference of 27 and 3 is 27 - 3, which can be expressed as (27 - 3). Similarly, the difference of 16 and 4 is 16 - 4, which can be expressed as (16 - 4).

Now, we need to divide the difference of 27 and 3 by the difference of 16 and 4. We can use the division operator (/) to represent the division operation.

Therefore, the expression that represents the given statement is:

(27 - 3) / (16 - 4)

Simplifying this expression further, we have:

24 / 12

The difference of 27 and 3 is 24, and the difference of 16 and 4 is 12. So, the expression simplifies to:

2

Hence, the expression (27 - 3) / (16 - 4) is equivalent to 2.

In summary, the expression that represents the statement "Divide the difference of 27 and 3 by the difference of 16 and 4" is (27 - 3) / (16 - 4), which simplifies to 2.

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Answer:

the radius of the cylinder is approximately 2.26 feet.

Step-by-step explanation:

To find the radius of the cylinder, we can use the formula for the volume of a cylinder:

Volume = π * radius^2 * height

Given that the volume is 400 feet, the height is 25 feet, and using π ≈ 3.14, we can rearrange the formula to solve for the radius:

400 = 3.14 * radius^2 * 25

Divide both sides of the equation by (3.14 * 25):

400 / (3.14 * 25) = radius^2

Simplifying:

400 / 78.5 ≈ radius^2

5.09 ≈ radius^2

To find the radius, we take the square root of both sides:

√5.09 ≈ √(radius^2)

2.26 ≈ radius

Rounding to the nearest hundredth, the radius of the cylinder is approximately 2.26 feet.

Answer:

Step-by-step explanation:

Volume Formula for a cylinder is  V=πr²h

Substitute the following:    400 = 3.14(r²)(25)

r=[tex]\sqrt{\frac{V}{\pi h} }[/tex]

r=[tex]\sqrt{\frac{400}{\pi 25} }[/tex]

r≈2.25676ft

For a collection of numbers to be considered a sequence, it is essential that they have a law of formation, are ordered in a specific manner, and can be either finite or infinite.

B: They have a law of formation:

A sequence is a set of numbers arranged in a specific order according to a rule or pattern. The numbers in a sequence are not random but follow a specific law of formation.

This law can be a mathematical formula, a recursive relationship, or any other systematic pattern that determines the values of the sequence. Without a well-defined law of formation, a collection of numbers cannot be considered a sequence.

C: They are ordered:

In a sequence, the numbers are arranged in a specific order or sequence. The order of the numbers is crucial and defines the pattern and structure of the sequence.

Each number in the sequence has a unique position or index that determines its place in the sequence. The order of the numbers allows us to identify the next number or predict the pattern of the sequence. Without the concept of order, the numbers would simply be a set of unrelated elements and not a sequence.

A: They may or may not be infinite:

Sequences can be finite or infinite. A finite sequence has a specific number of terms, and once the pattern or rule is established, the sequence ends.

On the other hand, an infinite sequence continues indefinitely, and its terms extend infinitely in one direction or both directions. Whether a sequence is finite or infinite depends on the context and the specific rule or pattern that governs its formation.

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Note: the translated question is:

To have a sequence it is essential that the numbers that form it:

A: be infinite

B: they have a law of formation C: they are ordered

Answer:

x intercept : -1

y intercept : 3

Step-by-step explanation:

We have 3x - y = -3 ---eq(1)

The x intercept is the value of x when y = 0 in eq(1),

⇒ 3x - 0 = -3

⇒ x = -3/3

⇒ x = -1

The y intercept is the value of y when x = 0 in eq(1),

⇒ 3(0) - y = -3

⇒ -y = -3

⇒ y = 3

Answer:

Let's denote:

- A as the event "student is accepted at FSU"

- B as the event "student is accepted at UF"

We have the following probabilities given:

- P(A) = 0.4 (probability of being accepted at FSU)

- P(B|A) = 0.6 (probability of being accepted at UF given acceptance at FSU)

- P(B'|A') = 0.9 (probability of not being accepted at UF given non-acceptance at FSU), where B' is the complement of B (not being accepted at UF) and A' is the complement of A (not being accepted at FSU).

We want to find P(A|B'), or the probability of being accepted at FSU given non-acceptance at UF.

To calculate this, we can use Bayes' Theorem, which states that:

P(A|B') = P(B'|A) * P(A) / P(B')

We can calculate P(B'|A), the probability of non-acceptance at UF given acceptance at FSU, by using the fact that the sum of probabilities of complementary events equals 1:

P(B'|A) = 1 - P(B|A) = 1 - 0.6 = 0.4

Next, we need to find P(B'), the total probability of non-acceptance at UF. We can use the law of total probability to calculate this:

P(B') = P(B'|A) * P(A) + P(B'|A') * P(A')

P(B') = 0.4 * 0.4 + 0.9 * 0.6 = 0.16 + 0.54 = 0.7

Finally, we can substitute these values into Bayes' theorem:

P(A|B') = P(B'|A) * P(A) / P(B') = 0.4 * 0.4 / 0.7 ≈ 0.229

Therefore, the probability that a student is accepted at FSU given that they are not accepted at UF is approximately 22.9%.

Answer:

Step-by-step explanation:

Let's break down the given information step by step to solve the problem:

The sum of three numbers x, y, and z is 165: x + y + z = 165.

When the smallest number x is multiplied by 7, the result is n: 7x = n.

The value n is obtained by subtracting 9 from the largest number y: y - 9 = n.

The value n is also obtained by adding 9 to the third number z: z + 9 = n.

We can use this information to form a system of equations:

Equation 1: x + y + z = 165

Equation 2: 7x = n

Equation 3: y - 9 = n

Equation 4: z + 9 = n

To find the product of the three numbers, we need to determine the values of x, y, z, and n.

First, let's solve for n using Equation 2:

7x = n

Now, let's substitute the value of n into Equations 3 and 4:

Equation 3: y - 9 = 7x

Equation 4: z + 9 = 7x

We can rearrange Equation 3 to express y in terms of x:

y = 7x + 9

Now, we can substitute the value of y in Equation 1:

x + (7x + 9) + z = 165

Simplifying:

8x + z = 156

Now, we have two equations:

Equation 4: z + 9 = 7x

8x + z = 156

We can solve this system of equations to find the values of x and z:

From Equation 4, we have z = 7x - 9.

Substituting z in Equation 8x + z = 156:

8x + (7x - 9) = 156

15x - 9 = 156

15x = 165

x = 11

Substituting x = 11 in Equation 4:

z + 9 = 7(11)

z + 9 = 77

z = 68

Now, we have the values of x = 11, y = 7x + 9 = 7(11) + 9 = 86, and z = 68.

The product of the three numbers x, y, and z is:

Product = x * y * z = 11 * 86 * 68 = 64648.

Therefore, the product of the three numbers is 64648.

Answer: D. 145°

Step-by-step explanation:

Since it is a parallelogram given by the symbol, then angle B is equal to angle D which is 145°.

Answer:

the effective tax rate for a taxable income of $175,000 is approximately 21.02%.

Step-by-step explanation:

Let's break down the income into the corresponding tax brackets:

The first $10,275 is taxed at a rate of 10%.

Tax on this portion: $10,275 * 0.10 = $1,027.50

The income between $10,276 and $41,175 is taxed at a rate of 12%.

Tax on this portion: ($41,175 - $10,276) * 0.12 = $3,710.88

The income between $41,176 and $89,075 is taxed at a rate of 22%.

Tax on this portion: ($89,075 - $41,176) * 0.22 = $10,656.98

The income between $89,076 and $170,050 is taxed at a rate of 24%.

Tax on this portion: ($170,050 - $89,076) * 0.24 = $19,862.88

The income between $170,051 and $175,000 is taxed at a rate of 32%.

Tax on this portion: ($175,000 - $170,051) * 0.32 = $1,577.44

Now, sum up all the taxes paid:

$1,027.50 + $3,710.88 + $10,656.98 + $19,862.88 + $1,577.44 = $36,836.68

The effective tax rate is calculated by dividing the total tax paid by the taxable income:

Effective tax rate = Total tax paid / Taxable income

Effective tax rate = $36,836.68 / $175,000 = 0.21024 (rounded to the nearest hundredth)

Answer:

  19 or 26

Step-by-step explanation:

You want the value of 3n² -8n -9, given that n(n -3) = 10.

We recognize that 10 = 5·2 and that these factors differ by 3. This means n(n -3) = 10 is equivalent to saying n ∈ {-2, 5}.

The value of 3n² -8n -9 will be one of ...

  (3n -8)n -9 = (3(-2) -8)(-2) -9 = (-14)(-2) -9 = 19 . . . . for n = -2

or

  (3(5) -8)(5) -9 = (7)(5) -9 = 26 . . . . . . . for n = 5

The expression 3n² -8n -9 will be either 19 or 26.

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Answer:

Step-by-step explanation:

Converting between Fraction, Decimal, and Percent:

Fraction to Decimal: Divide the numerator by the denominator. The result is the decimal form.

Example: 1/4 = 1 ÷ 4 = 0.25

Decimal to Fraction: Write the decimal as a fraction by placing the decimal value over the appropriate power of 10.

Example: 0.75 = 75/100 = 3/4

Fraction to Percent: Divide the numerator by the denominator and multiply by 100.

Example: 1/3 = (1 ÷ 3) × 100 = 33.33...%

Percent to Fraction: Write the percent as a fraction with a denominator of 100 and simplify if necessary.

Example: 40% = 40/100 = 2/5

Decimal to Percent: Multiply the decimal by 100 and add the percent symbol (%).

Example: 0.75 = 0.75 × 100 = 75%

Percent to Decimal: Divide the percent by 100.

Example: 40% = 40 ÷ 100 = 0.4

Alonso spent all of his money, this confirms that he can buy 3 bags of avocados.

Alonso has $21.00 to spend on eggs and avocados. He buys eggs that cost $2.50, which leaves him with $18.50. Since the store only sells avocados in bags of 3, he will need to find the cost per bag in order to calculate how many bags he can buy.

First, divide the cost of 3 avocados by 3 to find the cost per avocado. $5.00 ÷ 3 = $1.67 per avocado.

Next, divide the money Alonso has left by the cost per avocado to find how many avocados he can buy.

$18.50 ÷ $1.67 per avocado = 11.08 avocados.

Since avocados only come in bags of 3, Alonso needs to round down to the nearest whole bag. He can buy 11 avocados, which is 3.67 bags.

Thus, he will buy 3 bags of avocados.Let's test our answer to make sure that Alonso has spent all his money:

$2.50 for eggs3 bags of avocados for $5.00 per bag, which is 9 bags of avocados altogether. 9 bags × $5.00 per bag = $45.00 spent on avocados.

Total spent:

$2.50 + $45.00 = $47.50

Total money had:

$21.00

Remaining money:

$0.00

Since Alonso spent all of his money, this confirms that he can buy 3 bags of avocados.

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Answer:

Step-by-step explanation:

Let's calculate the expression step by step:

93 - (15 × 10) + (160 ÷ 16)

First, we perform the multiplication:

93 - 150 + (160 ÷ 16)

Next, we perform the division:

93 - 150 + 10

Finally, we perform the subtraction and addition:

-57 + 10

The result is:

-47

Therefore, 93 - (15 × 10) + (160 ÷ 16) equals -47.

a. The absolute maximum of g is 4.

The absolute minimum of g is -4.

b. The absolute maximum of h is 3.

The absolute minimum of h is -4.

In Mathematics and Geometry, the vertical asymptote of a function simply refers to the value of x (x-value) which makes its denominator equal to zero (0).

By critically observing the graph of the polynomial function g shown above, we can logically deduce that its vertical asymptote is at x = 3. Furthermore, the absolute maximum of the polynomial function g is 4 while the absolute minimum of g is -4.

In conclusion, the absolute maximum of the polynomial function h is 3 while the absolute minimum of h is -4.

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Answer: 2,2

Step-by-step explanation:

trust me

Answer:

Step-by-step explanation:

To solve the inequality -5 < 2x - 1 < 3, we will break it down into two separate inequalities and solve each one individually.

First, let's solve the left inequality:

-5 < 2x - 1

Add 1 to both sides:

-5 + 1 < 2x - 1 + 1

-4 < 2x

Divide both sides by 2 (remembering to reverse the inequality when dividing by a negative number):

-4/2 < 2x/2

-2 < x

Now, let's solve the right inequality:

2x - 1 < 3

Add 1 to both sides:

2x - 1 + 1 < 3 + 1

2x < 4

Divide both sides by 2:

2x/2 < 4/2

x < 2

So, the solutions to the inequalities are:

-2 < x < 2

This means that x is greater than -2 and less than 2.

Answer:

R = {(0, 8), (0, -8), (8, 0), (-8, 0), (6, ±2), (-6, ±2), (2, ±6), (-2, ±6)}

Step-by-step explanation:

Since [tex](\pm8)^2+0^2=64[/tex], [tex]0^2+(\pm 8)^2=64[/tex], [tex](\pm 6)^2+2^2=64[/tex], and [tex]6^2+(\pm 2)^2=64[/tex], then those are your integer solutions to find R.

The value of cos E in the triangle is √2/2

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. Trigonometric ratio is only applied in right triangles.

Some of the trigonometric functions are ;

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

In the triangle taking reference from angle E, 5 is the opposite and 5 is the adjascent and 5√2 is the hypotenuse.

Therefore;

cos E = 5/5√2

cos E = 1/√2

rationalizing the value;

cos E = √2/2

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The function for which f(x) is equal to f⁻¹(x) is: C. [tex]f(x) = \frac{x+1}{x-1}[/tex]

In this exercise, you are required to determine the inverse of the function f(x) with an equivalent inverse function f⁻¹(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

[tex]f(x) = y = \frac{x+6}{x-6} \\\\x=\frac{y+6}{y-6}[/tex]

x(y - 6) = y + 6

y = xy - 6x - 6

f⁻¹(x) = (-6x - 6)/(x - 1) ⇒ Not equal.

Option B.

[tex]f(x) = y = \frac{x+2}{x-2} \\\\x=\frac{y+2}{y-2}[/tex]

x(y - 2) = y + 2

y = xy - 2x - 2

f⁻¹(x) = (-2x - 2)/(x - 1) ⇒ Not equal.

Option C.

[tex]f(x) = y = \frac{x+1}{x-1} \\\\x=\frac{y+1}{y-1}[/tex]

x(y - 1) = y + 1

y - xy = x + 1

f⁻¹(x) = (x + 1)/(x - 1) ⇒ equal.

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Answer:

64 and 65

Step-by-step explanation:

Let the two lockers be x and (x+1) since they are consecutive

The sum is 129 so,

x + (x+1) = 129

2x + 1 = 129

2x = 128

x = 64

x + 1 = 65

The locker numbers are 64 and 65

The completed sequence would then be: 2, 4, 7, 9, 16, 19, 29.

To complete the given number sequence, let's analyze the pattern and identify the missing terms.

Looking at the given sequence 2, 4, 7, __, 16, __, 29, __, we can observe the following pattern:

The difference between consecutive terms in the sequence is increasing by 1. In other words, the sequence is formed by adding 2 to the previous term, then adding 3, then adding 4, and so on.

Using this pattern, we can determine the missing terms as follows:

To obtain the third term, we add 2 to the second term:

7 + 2 = 9

To find the fifth term, we add 3 to the fourth term:

16 + 3 = 19

To determine the seventh term, we add 4 to the sixth term:

__ + 4 = 23

Therefore, the missing terms in the sequence are 9, 19, and 23.

By identifying the pattern of increasing differences, we can extend the sequence and fill in the missing terms accordingly.

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The value of x is 5625 when √x + √y=138 and x-y=1656.

To solve for x using the given equations, we will use the method of substitution. Let's go through the steps:

Start with the equation √x + √y = 138.

We want to express y in terms of x, so solve for y in terms of x by isolating √y:

√y = 138 - √x

Square both sides of the equation to eliminate the square root:

(√y)² = (138 - √x)²

y = (138 - √x)²

Now, substitute the expression for y in the second equation x - y = 1656:

x - (138 - √x)² = 1656

Expand the squared term:

x - (138² - 2 * 138 * √x + (√x)²) = 1656

Simplify the equation further:

x - (19044 - 276 * √x + x) = 1656

Combine like terms and rearrange the equation:

-19044 + 276 * √x = 1656

Move the constant term to the other side:

276 * √x = 20700

Divide both sides of the equation by 276 to solve for √x:

√x = 20700 / 276

√x = 75

Square both sides to solve for x:

x = (√x)²

x = 75²

x = 5625

Therefore, the value of x is 5625.

By substituting this value of x back into the original equations, we can verify that √x + √y = 138 and x - y = 1656 hold true.

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