Can anyone help me with this?

The partial derivative of function f with respect to x at point (1, 1, 4) is 7, the partial derivative with respect to y is 3, and the partial derivative with respect to z is -16.

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The coordinate vector for w relative to the basis S is [w]s = (16, -10).

To find the coordinate vector for w relative to the basis S = {u1, u2} for R2 with u1 = (1,2), u2 = (1,3), and w = (6,2), we need to solve for the scalars a and b such that w = a*u1 + b*u2.

Step 1: Write the linear combination equation:
w = a*u1 + b*u2
(6,2) = a*(1,2) + b*(1,3)

Step 2: Write the system of equations for the components of the vectors:
6 = a + b
2 = 2a + 3b

Step 3: Solve the system of equations. We can eliminate a by multiplying the first equation by 2 and then subtracting the second equation:
12 = 2a + 2b
2 = 2a + 3b
----------------
10 = -b

So, b = -10.

Step 4: Plug b back into the first equation to find a:
6 = a + (-10)
a = 16

Step 5: Write the coordinate vector [w]s:
[w]s = (a, b) = (16, -10)

So, the coordinate vector for w relative to the basis S = {u1, u2} for R2 with u1 = (1,2), u2 = (1,3), and w = (6,2) is [w]s = (16, -10).

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The probability that Gareth selected a yellow tile or a tile with an odd number is 9/12 or 3/4.

To solve this question, we need to first determine the total number of tiles in the set and then count the number of yellow tiles and the number of tiles with odd numbers.
Looking at the set of tiles, we can see that there are a total of 12 tiles. To count the number of yellow tiles, we can see that there are 4 yellow tiles. To count the number of tiles with odd numbers, we can see that there are 6 tiles with odd numbers (1, 3, 5, 7, 9, and 11).
Now that we have these counts, we can use the formula for the probability of the union of two events to find the probability that Gareth selected a yellow tile or a tile with an odd number. The formula for the probability of the union of two events is:
P(A or B) = P(A) + P(B) - P(A and B)
where A and B are two events, P(A) is the probability of event A, P(B) is the probability of event B, and P(A and B) is the probability of both events happening together.
In this case, A is the event that Gareth selected a yellow tile and B is the event that he selected a tile with an odd number. So we have:
P(yellow or odd) = P(yellow) + P(odd) - P(yellow and odd)
Plugging in the values we counted earlier, we get
P(yellow or odd) = 4/12 + 6/12 - 1/12
Simplifying this expression, we get:
P(yellow or odd) = 9/12
Therefore, the probability that Gareth selected a yellow tile or a tile with an odd number is 9/12 or 3/4.

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Answer:

It will take the snail 10 minutes to travel 25 feet.

Step-by-step explanation:

15 feet / 6 minutes = 25 feet / x minutes

15 feet * x = 6 minutes * 25 feet

15x = 150

x = 10

Answer: 2

Step-by-step explanation:

Answer:

2

Step-by-step explanation:

1 = the first number in numbers after 0, of course now... if you add the first number plus the first number 1 + 1 you should end up with 2

Hope this helps L M A O

Explain how to write the exponential model, and then write the model: B. substitute the values of x and y from one of the points into the equation and solve for a. The equation is [tex]y = 7.39a^x[/tex]

How can you use the exponential model to find the value of y when x = 8: D. substitute 8 for x in the model and simplify. Thus, y = 22029.

In Mathematics, an exponential function can be modeled by using the following mathematical equation:

[tex]f(x) = a(b)^x[/tex]

Where:

By substituting the values of x and y from one of the points into the equation, we have:

54.61 = ae²

a = 54.61/e²

a = 7.39

Therefore, the exponential function is [tex]y = 7.39a^x[/tex].

When x = 8, we have;

[tex]y = f(8) = 7.39a^8[/tex]

y = f(8) = 22029.28 ≈ 22029.

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For 9.9: To prove the first statement, it is assumed that S CT, and then it is shown that 25 C 2T. To prove the second statement, it is assumed that 25 C 2T, and then it is shown that S CT.

For 9.10: The statement "23 n 21 = 2SNT" is proved by constructing a bijection between the set of ordered pairs (A, B) where A is a 2-element subset of {1, 2, 3} and B is a 1-element subset of {1, 2}. The bijection is defined by the function f(A, B) = (x, y, z), where x is the largest element of A, y is the unique element of B, and z is the remaining element of {1, 2, 3} - A - {y}

For 9.9: To prove that S CT if and only if 25 C 2T, we need to show two things: 1. If S CT, then 25 C 2T. 2. If 25 C 2T, then S CT.

To prove the first statement, assume S CT. This means that every element of S is also an element of T (i.e., S is a subset of T). We want to show that 25 C 2T, which means that every 2-element subset of 25 is also a subset of T.

Let A be an arbitrary 2-element subset of 25. Since A has exactly two elements, we can write it as A = {a, b}, where a and b are distinct elements of 25. Since 25 C T, we know that a and b are both elements of T. Therefore, A is a subset of T, and we have shown that 25 C 2T.

To prove the second statement, assume 25 C 2T. This means that every 2-element subset of 25 is also a subset of T. We want to show that S CT, which means that every element of S is also an element of T.

Let x be an arbitrary element of S. We want to show that x is also an element of T. Since 25 C 2T, we know that {x, y} is a subset of T for every y in S (i.e., every pair of elements from S is also a pair of elements from T). In particular, this means that {x, x} (i.e., the set containing only x) is a subset of T. But this means that x is an element of T, since every subset of {x} is also a subset of {x, x}.

Therefore, we have shown both directions of the "if and only if" statement, and we can conclude that S CT if and only if 25 C 2T.

For 9.10: To prove that 23 n 21 = 2SNT, we need to show that there is a bijection between the set of ordered pairs (A, B) where A is a 2-element subset of {1, 2, 3} and B is a 1-element subset of {1, 2}, and the set SNT (i.e., the set of all ordered triples (x, y, z) where x is an element of {1, 2, 3}, y is an element of {1, 2}, and z is an element of {1, 2, 3} - {x} - {y}).

To construct the bijection, we will define a function f from the set of ordered pairs (A, B) to SNT, and show that f is both injective (i.e., no two different ordered pairs map to the same element of SNT) and surjective (i.e., every element of SNT is the image of some ordered pair under f).

Let (A, B) be an arbitrary ordered pair in the domain of f. We will define f(A, B) to be the ordered triple (x, y, z), where x is the largest element of A, y is the unique element of B, and z is the remaining element of {1, 2, 3} - A - {y}.

To see that f is injective, suppose that (A, B) and (A', B') are two different ordered pairs in the domain of f such that f(A, B) = f(A', B'). This means that x = x', y = y', and z = z', where x and x' are the largest elements of A and A', respectively, y and y' are the unique elements of B and B', respectively, and z and z' are the remaining elements of {1, 2, 3} - A - {y} and {1, 2, 3} - A' - {y'}, respectively.

Since x and x' are both elements of A and A', respectively, and A and A' are both 2-element subsets of {1, 2, 3}, we must have x = x'. But then y = y' and z = z', since these are uniquely determined by x and the choice of A or A'. Therefore, we have (A, B) = (A', B'), and f is injective.

Since x is an element of {1, 2, 3}, we can write {1, 2, 3} - {x} = {a, b} for some distinct elements a and b of {1, 2, 3}. Let A = {x, a}, which is a 2-element subset of {1, 2, 3}. Since y is an element of {1, 2}, we can write {1, 2} - {y} = {c} for some element c of {1, 2}. Let B = {y}, which is a 1-element subset of {1, 2}.

Therefore, since f is both injective and surjective, it is a bijection between the set of ordered pairs (A, B) and the set SNT, and we have shown that 23 n 21 = 2SNT.

1. S ⊆ T, if and only if, some condition related to 25 and 2T.
2. A relationship involving intersection (denoted by ∩) and union (denoted by ∪) between sets S and T, possibly involving their complements (denoted by S' and T').

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Each term of the sequence is obtained by multiplying the previous term by 2/3.

To find the sequence represented by a generating function, we can expand the function as a power series and read off the coefficients. In this case, we can write:

1/(1-7z+10z²) = (1/[(1-2z)(1-5z)]) = (A/(1-2z)) + (B/(1-5z))

where A and B are constants we need to determine. We can use partial fraction decomposition to find A and B, which gives:

A = 1/3 and B = -1/3

Now we can express the generating function as:

1/(1-7z+10z²) = (1/3)/(1-2z) - (1/3)/(1-5z)

Next, we use the geometric series formula to expand each term as a power series:

(1/3)/(1-2z) = 1/3 + 2z/9 + 4z²/27 + 8z³/81 + ...

-(1/3)/(1-5z) = -1/3 - 5z/9 - 25z²/27 - 125z³/81 - ...

Finally, we can combine the coefficients of each power of z to obtain the sequence represented by the generating function. We have:

1/3, 2/9, 4/27, 8/81, ...

This is a geometric sequence with first term 1/3 and common ratio 2/3. Therefore, the nth term of the sequence is given by the closed-form expression:

aₙ = (1/3) * (2/3)ⁿ⁻¹

We can also express the sequence recursively by noting that:

a₁ = 1/3 aₙ = (2/3) * aₙ₋₁

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Answer:

(x - 3)(x + 1) , (x + 3)(x + 4)

Step-by-step explanation:

(a)

y = x² - 2x - 3

consider the factors of the constant term (- 3) which sum to give the coefficient of the x0- term (- 2)

the factors are - 3 and + 1 , since

- 3 × + 1 = - 3 and - 3 + 1 = - 2 , then

x² - 2x - 3 = (x - 3)(x + 1)

(b)

y = x² + 7x + 12

consider the factors of the constant term (+ 12) which sum to give the coefficient of the x- term (+ 7)

the factors are + 3 and + 4 , since

3 × 4 = + 12 and 3 + 4 = + 7 , then

x² + 7x + 12 = (x + 3)(x + 4)

It will take Lucas 2 hours to reach sea level.

First, let's define the terms:
Let d represent the distance (in meters) that Lucas needs to travel to reach sea level.
Let r represent the rate (in meters per minute) at which Lucas is driving down the mountain.
Let t represent the time (in minutes) it takes for Lucas to get to sea level.
We're given that Lucas is driving at a rate of 12 meters per minute (r = 12) and the sea level is 1,440 meters away (d = 1,440).

We need to find the time it takes (t) in hours.
We can write an equation to model the scenario using the formula:

distance = rate × time, or d = rt.

Plugging in the given values, we have:
1,440 = 12 × t
Now, we need to solve for t:
t = 1,440 / 12
t = 120 minutes
To convert this time into hours, we'll divide by 60 (since there are 60 minutes in an hour):
t (hours) = 120 minutes / 60
t (hours) = 2 hours.

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Answer:

First, we can simplify the left side of the equation by distributing 1/4:

1/4 (x - 2/5) = 1/4 x - 1/5

Now the equation becomes:

1/4 x - 1/5 = -1 1/2

We can convert the mixed number on the right side to an improper fraction:

-1 1/2 = -3/2

Adding 1/5 to both sides, we get:

1/4 x = -3/2 + 1/5

Combining the fractions on the right side:

1/4 x = -13/10

Multiplying both sides by 4:

x = -13/10 * 4 = -26/5

We can convert this to a mixed number in simplest form by dividing the numerator by the denominator:

x = -5 1/5

To ensure that the function f(x) is continuous at every 'x', we need to make sure it's continuous at the point x = 3. For a function to be continuous, the left-hand limit, right-hand limit, and the function value at the point should all be equal. Let's evaluate the limits:

1. Left-hand limit (x < 3): lim(x->3-) f(x) = lim(x->3-) x^2 = 3^2 = 9
2. Right-hand limit (x ≥ 3): lim(x->3+) f(x) = lim(x->3+) 2ax = 2a(3) = 6a
3. Function value at x = 3: f(3) = 2a(3) = 6a

For f(x) to be continuous, the left-hand limit, right-hand limit, and function value at x = 3 must be equal:

9 = 6a

To solve for 'a', divide both sides by 6:

a = 9/6 = 3/2

So, the value of 'a' that makes f(x) continuous at every 'x' is 3/2.

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Dominic should rent a house given his circumstances and his plans to study in another school.

A down payment is a sum of money that a buyer pays up front to purchase a significant item, like a house or a car. When purchasing a home, the down payment represents normally a portion of the entire cost, with the remaining balance being financed by a mortgage. Many elements, including the buyer's credit score, the kind of mortgage, and the lender's requirements, might affect the down payment amount.

Dominic wants to carry on his education by enrolling in a graduate programme at a different university in a different state. When deciding to settle down for an extended amount of time, buying a home usually makes more financial sense because buying and selling a home can be expensive.

The total savings are:

1500 + 3200 + 1000  

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According to the given question the nearest cent, of the sales tax on Barry's order is 676 cents.

The act of buying something for use in the manufacture of another good or service, or to sell again, is known as purchasing in business. Any organization's purchasing department plays a crucial role, and its significance shouldn't be understated.

To find the sales tax on Barry's order, we need to first calculate the total cost of his purchase, including the sales tax.

To find the sales tax on Barry's order in cents, we can follow the same process as before, but we need to convert the dollar amount to cents:

The cost of the wallet, sweater, and watch is $8.99 + $14.99 + $72.49 = $96.47, which is 9647 cents.

To find the sales tax, we can multiply the total cost by the tax rate, which is 7% or 0.07:

Sales tax = 9647 x 0.07 = 676.29

Rounding to the nearest cent, the sales tax on Barry's order is 676 cents.

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The cubic feet of concrete needed to pour a block with dimensions 3' (L) x 3' (W) x 3' (D) is 27 cubic feet.

It is given that a concrete block is 3 feet long, 3 feet wide, and 3 feet deep. In order to calculate the number of cubic feet of concrete is required, we will need to calculate the volume of the block.

1. Multiply the length (L), width (W), and depth (D) together to determine the volume

Volume = 3' x 3' x 3'

2. The result will give you the total cubic feet (C) of concrete required:

Volume = 27 cubic feet

So, you need 27 cubic feet of concrete to pour a block with the dimensions 3' (L) x 3' (W) x 3' (D).

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The true statement about qualitative design is "qualitative researchers regularly prepare a complicated array of records from a variety of sources."

Qualitative research usually includes accumulating records from multiple sources, along with interviews, observations, and files, among others. Researchers analyze those records in-depth to identify themes, styles, and relationships amongst them, frequently the use of an iterative technique to refine their expertise.

at the same time as qualitative research can offer valuable insights into complex phenomena, it isn't commonly designed to permit causal inferences, as it frequently involves small samples and is centered on exploring the reviews and views of contributors in place of testing hypotheses or making predictions.

Qualitative research may additionally contain some level of assessment, however that is commonly not pre-planned and can be extra exploratory in nature. in addition, reaching constancy of conditions in terms of research settings isn't always normally a number one goal of qualitative studies.

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Like your other problem just subtract the triangle from the pie piece that is created from the radii.

Area: [tex]\frac{80}{360}*\pi*6^2-\frac{1}{2}*6*6=7.1[/tex]

Q1) The ball experiences a pressure of 392,000 Pa at the bottom of the lake.

Q2) The ball is made of a dense material like lead or depleted uranium.

Q1) To calculate the pressure experienced by the ball at the bottom of the lake, we can use the formula

P = ρgh

Where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the depth of the ball in the water.

We know that the ball sinks 40 m in 3.6 s, so we can calculate its velocity using the formula

v = gt

Where v is the velocity, g is the acceleration due to gravity (9.8 m/s^2), and t is the time (3.6 s).

v = gt = 9.8 m/s² × 3.6 s = 35.28 m/s

Using the formula for the depth of an object in water

h = (ρ_ball/ρ_water) × r

Where ρ_ball is the density of the ball, ρ_water is the density of water (1000 kg/m³), and r is the radius of the ball.

We can rearrange this formula to solve for the density of the ball:

ρ_ball = (h/ r) × ρ_water

ρ_ball = (40 m / 0.12 m) × 1000 kg/m³ = 333,333.33 kg/m³

This is much higher than the density of any common metal, so it's possible that the ball is made of a dense material like lead or depleted uranium.

Q2) Plugging the values into the pressure formula

P = ρgh = 1000 kg/m³ × 9.8 m/s² × 40 m = 392,000 Pa

So, the ball experiences a pressure of 392,000 Pa at the bottom of the lake.  The density of the ball is calculated to be 333,333.33 kg/m³, which is much higher than the density of any common metal. It's possible that the ball is made of a dense material like lead or depleted uranium.

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The scatterplot of butter supply on the y-axis and happiness on the x-axis may or may not have the same shape and slope as another scatterplot with different variables. It depends on the correlation between butter supply and happiness, which can be determined by analyzing the scatterplot's shape and calculating its slope.

If the scatterplot has butter supply on the y-axis and happiness on the x-axis, the general shape and slope of the plot may or may not be the same, depending on the correlation between these two variables. Let's analyze the potential outcome step by step:
Data collection: First, we need to gather data on butter supply and happiness levels. This can be done by measuring the quantity of butter supply (in tons, for example) and some index for happiness (e.g., the World Happiness Index).
Plot the scatterplot: Next, we'll plot the data points with butter supply on the y-axis and happiness on the x-axis. Each point will represent a location or a group of people with specific butter supply and happiness levels.
Analyze the shape: After plotting the data points, we will observe the overall shape of the scatterplot. If there's a clear pattern, such as an upward or downward trend, this indicates a correlation between butter supply and happiness. If there's no discernible pattern, it means there's no significant correlation.
Determine the slope: If the scatterplot has a clear pattern, we can calculate the slope by fitting a regression line. The slope will indicate the relationship between butter supply and happiness. A positive slope indicates that as butter supply increases, happiness also increases, while a negative slope suggests the opposite.
In conclusion, the scatterplot of butter supply on the y-axis and happiness on the x-axis may or may not have the same shape and slope as another scatterplot with different variables. It depends on the correlation between butter supply and happiness, which can be determined by analyzing the scatterplot's shape and calculating its slope.

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A4 = [1 32/27 32/27 81]. To compute A4, we first need to find A. From the given equation A = PDP-1, we can substitute the values of P and D to get:

A = [1 2 2 3] [1 0 0 3]^-1

To find the inverse of D, we can simply take the reciprocal of each non-zero element on the main diagonal. In this case, the inverse of D is:

D^-1 = [1 0 0 1/3]

Substituting this value into the equation for A, we get:

A = [1 2 2 3] [1 0 0 1/3]
 = [1 2/3 2/3 3]

Now that we have A, we can easily compute A4 by raising A to the fourth power. That is:

A4 = A x A x A x A
  = [1 2/3 2/3 3] x [1 2/3 2/3 3] x [1 2/3 2/3 3] x [1 2/3 2/3 3]
  = [1 32/27 32/27 81]

Therefore, A4 = [1 32/27 32/27 81].

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The relative rate of change of f(x) at x = 19 can be found by taking the derivative of f(x) with respect to x and then evaluating it at x = 19. The derivative of f(x) = 188 - 2x is -2. Therefore, the relative rate of change of f(x) at x = 19 is -2.000.

Note: The negative sign indicates that f(x) is decreasing at x = 19.
To find the relative rate of change of f(x) at x=19, we first need to find the derivative of f(x) with respect to x, which represents the rate of change.

f(x) = 188 - 2x

The derivative, f'(x), is the derivative of the constant (188) minus the derivative of the term (-2x) with respect to x:

f'(x) = 0 - (-2) = 2

Now, we find the value of the derivative at x=19:

f'(19) = 2

To find the relative rate of change, we'll divide the rate of change (f'(19)) by the function value at x=19:

f(19) = 188 - 2(19) = 188 - 38 = 150

Relative rate of change = f'(19) / f(19) = 2 / 150 ≈ 0.013

So, the relative rate of change of f(x) at x = 19 is approximately 0.013 (rounded to three decimal places as needed).

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Answer:

y - 3 = -2(x + 4)

Step-by-step explanation:

For the given line, start at (-4, -3). Go up 4 units, then right 8 units. You will end at

(4, 1). The slope of this line is 1/2, so the slope of the perpendicular line will be -2 since the slopes of perpendicular lines are negative reciprocals of each other. The perpendicular line passes through

(-4, 3), so the correct equation is

y - 3 = -2(x + 4).

The height of the cliff is approximately 269 feet.

We will be using the tangent function to find the height of the cliff.
Draw a right triangle with the rowboat at one corner, the cliff's base as the adjacent side, and the cliff's height as the opposite side.

The angle of elevation (73.8 degrees) is between the rowboat and the cliff's base.
We are given the adjacent side (75 feet) and we need to find the opposite side (height of the cliff).

To do this, we will use the tangent function:
tan(angle) = opposite side / adjacent side
Plug in the given values:
tan(73.8) = height / 75
Solve for the height:
height = tan(73.8) * 75
Calculate the value:
height ≈ 269.4 feet
Round the height to the nearest foot:
height ≈ 269 feet.

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A small pitcher holds 64 ounces of tea, and since each glass holds 16 ounces, the number of glasses that can be filled from the pitcher is 4.

To see why, we can start by converting the capacity of the pitcher from cups to ounces. Since 1 cup is equivalent to 8 fluid ounces, the pitcher can hold

Next, we divide the total volume of the pitcher by the volume of each glass to find the number of glasses that can be filled. In this case,

Therefore, the small pitcher can fill 4 glasses of tea.

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In a system consisting of the cascade of two LTI (linear time-invariant) systems with frequency responses, the overall frequency response can be found by taking the product of the individual frequency responses.

To explain this concept more clearly, let's say that the first system has a frequency response H1(jw), and the second system has a frequency response H2(jw). The overall frequency response, H(jw), of the cascade system can be found by taking the product of the two individual frequency responses:

H(jw) = H1(jw) × H2(jw)

This means that the output of the first system is fed into the second system as its input, and the overall output of the cascade system is the output of the second system.

It's worth noting that LTI systems are important in signal processing because they have properties that allow us to analyze and manipulate signals easily. For example, LTI systems have the property of linearity, which means that if we input a scaled version of a signal, the output will also be scaled by the same factor.

Additionally, LTI systems have the property of time-invariance, which means that the system's behavior doesn't change over time. These properties make LTI systems useful for a wide range of applications in signal processing and control.

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The mathematically appropriate way to say 1.27 is "one and twenty-seven hundredths," option E is the correct answer.

When expressing numbers with decimal places, it is important to use the appropriate mathematical terminology to ensure clarity and precision. The number 1.27 can be expressed as "one and twenty-seven hundredths."

The digits after the decimal point represent fractions of a whole number, with each digit to the right of the decimal point representing a power of ten that is one-tenth the value of the previous digit.

Using the correct terminology when discussing numbers with decimal places can help avoid confusion and miscommunication. In particular, expressing the fractional component using hundredths is often preferred when working with financial or scientific data.

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False. Since the domains of X, Y, and Z are the set of real negative numbers, the equation 22 - y2 = x must hold true for all real negative values of x and y.

VzVx3y (22 - y² = x), with the assumption that the domains of X, Y, and Z are the set of real negative numbers. Given this information, the statement is False.

Since the domains of X, Y, and Z are the set of real negative numbers, it means y² will always be positive (as squaring any negative number results in a positive number). Thus, the equation 22 - y² = x cannot hold true because 22 - y² will always be less than 22 (given y² is positive), making x a positive number, which contradicts the given domain of x being negative numbers.

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Answer: Yes 5=5

Step-by-step explanation:

If you plug the (X,Y) into the equation you'll get 5=3(4)-7

which is 5=12-7

5=5 and the equation is true

The restaurant should charge $17.50 if the ingredients cost $5.

The process of calculating the product of two or more numbers is known as multiplication. The multiplication of two numbers, say 'a' and 'b,' is written as 'a' multiplied by 'b. In mathematics, multiplication is defined as repeatedly adding a number with regard to another integer.

If a restaurant should charge its customers about 3.5 times the cost of the ingredients and the ingredients cost $5, then the restaurant should charge:

3.5 x $5 = $17.50

Therefore, the restaurant should charge $17.50 if the ingredients cost $5.

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Taking into account the definition of a system of linear equations, if the sum of two numbers is 30 and their product is 209, the number are 11 and 19.

Systems of linear equations are groupings of first degree equations with the same unknowns, of which a common solution must be found.

Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied.

In this case, a system of linear equations must be proposed taking into account that "x" and "y" are two numbers.

You know:

So, the system of equations to be solved is

x + y= 30

x×y=209

It is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.

In this case, isolating "x" from the first equation:

x= 30 - y

Substituting the expression in the second equation:

(30 - y)×y=209

Solving:

30y - y×y=209

30y - y² - 209= 0

Being this a quadratic function of the form ax² + by +c, then it can be solved by: [tex]x1,x2=\frac{-b+-\sqrt{b^{2}-4ac } }{2a}[/tex]

In this case, being a=-1, b=30 and c=-209, the equation is solved by:

[tex]y1=\frac{-30+\sqrt{30^{2}-4x(-1)x(-209) } }{2x(-1)}[/tex]

[tex]y1=\frac{-30+\sqrt{900-836 } }{-2}[/tex]

[tex]y1=\frac{-30+\sqrt{64} }{-2}[/tex]

[tex]y1=\frac{-30+8}{-2}[/tex]

[tex]y1=\frac{-22}{-2}[/tex]

y1= 11

Remembering that x= 30 - y, you get:

x1= 30 -11

x1= 19

and

[tex]y2=\frac{-30-\sqrt{30^{2}-4x(-1)x(-209) } }{2x(-1)}[/tex]

[tex]y2=\frac{-30-\sqrt{900-836 } }{-2}[/tex]

[tex]y2=\frac{-30-\sqrt{64} }{-2}[/tex]

[tex]y2=\frac{-30-8}{-2}[/tex]

[tex]y2=\frac{-38}{-2}[/tex]

y2= 19

Remembering that x= 30 - y, you get:

x2= 30 - 19

x2= 11

Finally, the number are 11 and 19.

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