Please which is the correct solution to the cubic equation.

​

The polynomial function:

f(x) = x³ - 125

Can be factorized into

f(x) = (x - 5)*(x² + 1*x + 5)

We want to factorize the polynomial expression:

f(x) = x³ - 125

And we want to factorize this in the following way:

f(x) = (x + a)*(x² + bx + c)

Trivially, 5 is a root of the function:

f(5) = 5³ - 125 = 0

Then we can write the function as:

f(x) = (x - 5)*(x² + bx + c)

And that must be equal to the original function:

(x - 5)*(x² + bx + c) = x³ - 125

Expanding the left side we will get:

x³ - 5x² + bx² + cx - 5bx - 5c = x³ - 125

x³ + (b - 5)x² + (c - 5b)x - 5c = x³ - 125

Then we can write the equations:

b - 5 = 0

c - 5b = 0

-c*5 = 125

Then we have:

c = 5

And with this equation:

c - 5b = 0

5 - 5b = 0

5 = 5b

5/5 = b

1  = b

Then the function can be written as:

f(x) = (x - 5)*(x² + 1*x + 5)

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The formula for for Rn is given as Rn = Σ 18/n (1 - 12i/n + 36i²/n²).

We are given the function f(x) = 3x² on the interval [-1, 5] . This gives

Δx =6/n and xi= -1 + 6i/n.

Therefore,

Rn =  Σ f(xi) Δx

     = Σ 3 (-1 + 6i/n)² 6/n

     = Σ 18/n (1 - 12i/n + 36i²/n²)

The limit of a function in mathematics is a key idea in calculus and analysis regarding the behavior of that function close to a specific input.

Informally, a function f gives each input x an output f(x). If f(x) approaches L as x approaches input p, we say that the function has a limit L at that location. More specifically, any input that is sufficiently close to p when f is applied forces the output value arbitrarily close to L.

The concept of limit is specifically used in the various definitions of continuity. Roughly speaking, a function is continuous if all of its limits agree with the values of the function. The term "limit" is also used in the definition of the mathematical term "derivative," which in one-variable calculus refers to the maximum value of the slope of secant lines on a function's graph.

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To find the domain of a function f, you need to identify the set of all possible input values for which the function produces a meaningful output. To find the range of a function f, you need to identify the set of all possible output values that the function produces when given a valid input.

The domain of a function f is the set of all possible input values for which the function produces a meaningful output, while the range is the set of all possible output values that the function produces when given a valid input.

To find the domain of a function f, you need to identify the set of all possible input values for which the function produces a meaningful output. To find the range of a function f, you need to identify the set of all possible output values that the function produces when given a valid input.

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An equivalent expression is an expression that is equal in value to a given expression. For example, consider the following expression: 4x + 2y

An equivalent expression is an expression that represents the same value as a given expression, regardless of the specific values of any variables that appear in the expression. Equivalent expressions can be obtained by applying the rules of algebra, such as the associative, commutative, and distributive properties.

This expression is equivalent to the following expressions:

2(2x + y)

2y + 4x

(4 + 2)x + 2y

All of these expressions are equivalent because they all represent the same value, regardless of the specific values of x and y. They can all be simplified to the same expression by applying the rules of the algebra.

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The 3rd graph has a relationships which is a function as a vertical can be drawn through any point and it only intersects once.

A function is a particular kind of relationship in which inputs are mapped to just one output. Consider a function as a machine for useful insight. The domain of the function, which describes the types of input that the machine will accept, states that it will produce one and only one output for each input.

A function's fundamental notation is of the form f ( x), where x is the input variable and f is the function's name. "F of X" should be read as this. Any time you see this notation, f(x) generally refers to a function's output, though these letters are subject to change.

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Answer:

Step-by-step explanation:

the 3rd option is the correct answer.

Answer: D. "none."

Step-by-step explanation: The complement of "at least one" is "none."

When we say "at least one," it means that there is a minimum of one of something. Therefore, the complement of "at least one" refers to the opposite condition, where there is no occurrence of that something.

Here's an example. Suppose you have a bag of marbles, and you are told that the bag contains at least one red marble. The complement of "at least one red marble" would mean that there are no red marbles in the bag.

All of the points from Table D which shows a proportional relationship has been plotted in the graph shown in the image attached below.

Generally speaking, the graph of any proportional relationship is represented by a straight line with the data points passing through the origin (0, 0). In Mathematics, a proportional relationship can be modeled by the following linear equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) for the data points contained in table D as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 1/2 = 3/6 = 4/8 = 2/10

Constant of proportionality, k = 1/2 or 0.5.

In conclusion, a linear equation for this proportional relationship can be correctly written and plotted using an online graphing calculator as follows:

y = kx

y = 0.5x

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The given lines are intersecting but not perpendicular.

What is intersecting lines?

In Euclidean geometry, the empty set, a point, or another line can intersect another line. For instance, computer graphics, motion planning, and collision detection can all benefit from differentiating these cases and locating the intersection.

Given:

          y = 6x - 2

2x - 12y = 24

⇒  6x - y = 2         ..(1)

2x - 12y = 24       ..(2)

Multiply second equation by 3,

6x - 36y = 72           ..(3)

Subtract equation (1) from equation (3).

     6x - 12y = 24

-    6x -    y  = 2

_______________

          -11y = 22

    y = -2

Plug y = -2 in equation (1)

6x - (-2) = 2

 6x + 2 = 2

       6x = 2 - 2

       6x = 0

         x = 0

The solution of the given system is (0, -2).

Hence, the given lines are intersecting but not perpendicular.

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(1) The number of pages in the math book rounded to the nearest hundreds will be 100 pages.

(2) The cost of the 10 DVDs will be $50.

When a number is rounded off, its value is maintained but is brought closer to the next number, simplifying the number. For entire numbers as well as decimals at different places of hundreds, tens, tenths, etc., it is done.

A number can be rounded off to its lower value if the number after the decimal is between 0 and 4. The number will be rounded off to its higher value if the number after it is between 5 and 9.

( 1 ) The number of pages in the math book will be 100, rounded to the nearest hundreds.

(2) The cost of the DVDs will be calculated as,

The cost of the 10 DVDs will be $50.

Cost = 10 x 4.95

Cost = $49.5

Cost = $50

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A scatter plot is a data visualization technique in data analysis that uses a graph to show the relationship between two variables. It is used to display the relationship between two variables in a two-dimensional graph, with one variable on the x-axis and the other variable on the y-axis.

Scatter plots are used to explore the relationship between two variables, identify any patterns, and detect outliers. Scatter plots can also be used to compare the trends between groups of data, such as the relationship between different age groups and salary, or the relationship between different countries and GDP.

Scatter plots can provide insights into which variables are most important when predicting a response variable. Scatter plots are a powerful tool for understanding relationships between variables and can be used in a wide variety of data analysis tasks.

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Step-by-step explanation:

Hello!

Point-slope form is this: y - y_{1} = m(x - x_{1})y−y

1

=m(x−x

1

) . In this form, m is the slope, x_{1}x

1

and y_{1}y

1

is an ordered pair, or point.

To find the slope, we need to use the slope formula. The slope formula is: \frac{y_{2}-y_{1} }{x_{2}-x_{1} }

x

2

−x

1

y

2

−y

1

. We can assign the point (-9, 6) to x_{1}x

1

and y_{1}y

1

and (-7, -8) to x_{2}x

2

and y_{2}y

2

.

Now, we can substitute the ordered pairs into the formula and solve.

m = \frac{-8 - 6}{-7 - (-9)} = \frac{-14}{2} = -7m=

−7−(−9)

−8−6

=

2

−14

=−7

Since the slope is equal to -7, we can substitute the slope and a point into the point-slope form.

Therefore, your point-slope formula is y - 6 + -7(x + 9) or y + 8 = -7(x + 7).

x - 9 > -15

Add 9 to both sides:
x - 9 + 9 > -15 + 9

x > -6

Option D would be your answer.

Hope this helps!

The correct answer is answer D (x > -6)

I just had this question and I got it right, Hope this helps. :)

The range of the given function g(x) is - 4 ≤ g(x) < 9.

        [tex]${\displaystyle G=\{(x,f(x))\mid x\in X\}.}[/tex]

Given is a graph of the figure as shown in the image.

The range of the given function g(x) is the set of all possible {y} values.

We can write the range of the function as -

- 4 ≤ g(x) < 9

Therefore, the range of the given function g(x) is - 4 ≤ g(x) < 9.

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Answer:

2

2Step-by-step explanation:

2

Answer:

The value of (3x + 2) when (7 + x = 5) is -4.

Step-by-step explanation:

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Answer:38

Step-by-step explanation: it is 180 degrease because it is spushed so (180-94)-50=38

Answer:

36⁰

Step-by-step explanation:

Angles in a triangle add upto 180⁰

94⁰+ 50⁰+b= 180⁰

144⁰+b=180⁰

b=180⁰-144⁰

b=36⁰

Using the given expression and applying the law of indices, The step by step solution gives the final answer to be 42.13 .

A mathematical expression is a combination of numbers, variables, and mathematical operations that defines a specific value or a set of values. Examples of mathematical expressions include 2+3, x^2+y^2, and sin(x) + cos(y). These expressions can be evaluated to find a specific numerical value or they can be simplified, rearranged or manipulated using algebraic rules.

The laws of indices, also known as the laws of exponents, are a set of rules that govern how exponents (or indices) are used in mathematical operations. These laws include:

These laws allow us to simplify expressions with exponents and make it easier to perform calculations involving them.

[tex]= 25 * \sqrt[401]2}^{302}\\= 25 * (2^{\frac {1}{401}})^{302}\\=25*2^{\frac {302}{401}}\\=25*2^{0.75}\\=42.13\\[/tex]

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Based on the use of substitution, The answer will be: (x, y) = (1, -2)

3x - 4y = 11

y + 3x = 1

Step 1: Solve one of the equations for one of the variables.

Let's solve the second equation for y:

y = -3x + 1

Step 2: Substitute this expression for y into the first equation:

3x - 4(-3x + 1) = 11

3x - (-12x + 4) = 11

3x + 12x = 15

15x = 15

x = 1

Step 3: Substitute this value of x back into one of the original equations to find the value of y:

y = -3(1) + 1 = -2

The solution is (x, y) = (1, -2)

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See transcribed text below

Solve the system using substitution.

3x-4y = 11 y + 3x = 1

([?], [_])

Enter

All the possible values of each expression is given as an inequality as;

3 < y - x < 6

We are given the inequalities of the domain and range as;

6 < x < 7

10 < y < 12

Thus, we can say that the maximum value of y-x will be the difference between the maximum value of y and the minimum value of x to give;

(y - x)max = 12 - 6

(y - x)max = 6

Similarly, minimum value of y - x will be the difference between the maximum value of x and the minimum value of y to give;

(y - x)min = 10 - 7

(y - x)min = 3

Thus, all possible values would be expressed as;

3 < y - x < 6

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Answer:

y = 7x

Step-by-step explanation:

I answered a similar question posted by you. The logic is the same so you should be able to figure it out

To graph y = 7x, use a graphing calculator.

If you wish to plot manually, take 2 values of x, find the y value using the equation and draw a straight line through those points

Point 1: When x = 0, y = 7 · 0 = 0. So (0,0) is one point

Point 2: When x = 1, y = 7; y = 7 · 1 = 7 So (7, 1) is the second point.

Draw a straight line through these points and you get your graph

The graph will be as shown.

[tex]\underset{same~base \qquad }{e^{4x}=e^{x^2+3}}\implies 4x=x^2+3\implies 0=x^2-4x+3 \\\\\\ 0=(x -3)(x - 1)\implies x= \begin{cases} 3\\ 1 \end{cases}[/tex]

The probability that the sum is a multiple of three is 0.2778

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event will not occur and 1 indicating that the event will definitely occur.

If a pair of six-sided dice is rolled, and the sum is recorded, then the possible sum is the numbers from (1 + 1) = 2 to (6 + 6) = 12.

From 1 to 12, the multiples of 3 are 3, 6, 9, and 12. These can be resulted from having the following dice rolled.

3 : 1 & 2, 2 & 1

6 : 1 & 5, 2 & 4, 3 & 3, 4 & 2, 5 & 1

9 : 3 & 6, 6 & 3

12 : 6 & 6

Since there are 10 possible outcomes wherein the sum is a multiple of 3, and there are (6 x 6) total outcomes, then the probability is

p = 10 / 36 = 0.2778

Hence, the probability that this sum is a multiple of three is 0.2778.

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Answer: hi :)

Step-by-step explanation: <3

The equation of variable is m= -9, x = -2, b = 10, w = 4. There was a final answer of solving this in equation.

Solving equation of m,

4m - 5 = - 41

4m =  - 41 + 5

4m = -36

m = -36 / 4

m = - 9.

Solving equation of x.

- 11 = 1 - 6x

-11 -1 = -6x

-12 = -6x

- 12/6 = x

x = -2

Solving equation of b.

4/5b + 5 = 13

4/5b = 13 - 5

4/5b = 8

b = 8 x 5/4

b = 40/4

b = 10

Solving equation of w.

- 1/2w - 8 = -6

-1/2w = -6 + 8

-1/2w = 2

w = 2 x 2/1

w = 4/1

w = 4

Hence. the answer is -9, -2, 10, 4.

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Loosen monetary policy causes interest rates to decrease and consumption and investment to increase.

When a central bank, such as the Federal Reserve in the United States, loosens monetary policy, it is typically trying to stimulate economic growth by making borrowing cheaper for individuals and businesses. One of the ways it can do this is by lowering interest rates.

When interest rates decrease, it becomes less expensive for individuals and businesses to borrow money, which can encourage them to spend more on things like homes, cars, and business expansion. This increased spending can lead to a boost in economic growth, as consumer and business spending are two major drivers of economic activity.

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Answer:

The answer for the first question is 24 medium pizzas, because:

5 times 15 = 75

291 - 75 = 216

216/9 = 24

The answer to the second question is 364.

39 + 52 = 91

91 times 4 = 364

Step-by-step explanation:

Hope it helps!

Answer:

Longer side= 36m, shorter side= 27m and hypotenuse=45m

Step-by-step explanation:

Let x be the longer leg of the triangle

According to the problem, the shorter leg of the triangle is 9 shorter than the longer leg, so the length of the shorter leg is x - 9

The hypotenuse is 9 longer than the longer leg, so the length of the hypotenuse is x + 9

We know that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So we can use the Pythagorean theorem:

(x + 9)^2 = x^2 + (x - 9)^2

Expanding and simplifying the equation:

x^2 + 18x + 81 = x^2 + x^2 - 18x + 81

x^2-36x=0

x=0 or, x=36

Since, x=0 is not possible in this case, we consider x=36 as the solution.

Thus, x=36, x-9=27 and x+9=45.

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The measure of the smallest angle in the diagram is given as follows:

29º.

The angle addition postulate states that if two angles share a common vertex and a common angle, forming a combination, the larger angle will be given by the sum of the smaller angles.

From the diagram presented at the end of the answer, we have that angle FLI is formed by a combination of angles FLG, GLH and HLI.

Then the value of x can be obtained as follows:

m < FLI = m < FLG + m < GLH + m < HLI

2x - 1 + x + 17 + 4x - 15 = 106

7x + 1 = 106

7x = 105

x = 105/7

x = 15.

Then the angle measures are given as follows:

Meaning that the measure of the smallest angle is of 29º.

The diagram is presented at the end of the answer.

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x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex] , y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex] The above mentioned is the solution of the given system of differential equation dx dt = 4x + 5y dy dt = 10x + 9y.

given  dx/dt = 4x+ 5y  and dy/dt = 10x + 9y

X'(t) = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]  X

where X = [tex]\left[\begin{array}{ccc}x\\y\\\end{array}\right][/tex]

so A = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]

now we need to find eigen value of the matrix and the corresponding eigen vector.

| A- λI | = 0

so after equation the value to zero

λ = 14 and λ = -1

now for the  λ = 14 corresponding eigen vector = [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]

for λ = -1 corresponding eigen vector =  [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

So general equation is given by:

X(t) = [tex]c_1e^1^4^t[/tex]  [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]  +  [tex]c_2e^-^t[/tex]   [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

after solving this

x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex]

y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex]

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