solve the equation for all values of x if x^2-12x+27=0
by completing the square

For the value of x = 4, the inequality equation that is true is option C: x/2 < 3.

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

The first inequality equation is - x + 5 < 3

Substitute the value of x = 4 in the equation.

4 + 5 < 3

Now, simplify the equation -

9 < 3

It is known that 9 is not less than 3, but greater than 3.

Therefore, this inequality is false.

The second inequality equation is - 9x > 36

Substitute the value of x = 4 in the equation.

9(4) > 36

Now, simplify the equation -

36 < 36

It is known that 36 is not less than 36, but equal 36.

Therefore, this inequality is false.

The third inequality equation is - x/2 < 3

Substitute the value of x = 4 in the equation.

4/2 < 3

Now, simplify the equation -

2 < 3

It is known that 2 is less than 3.

Therefore, this inequality is true.

The fourth inequality equation is - 18 < x + 8

Substitute the value of x = 4 in the equation.

18 < 4 + 8

Now, simplify the equation -

18 < 12

It is known that 18 is not less than 12, but greater than 12.

Therefore, this inequality is false.

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Which inequality is true when x = 4?

A. x + 5 < 3

B. 9x > 36

C. ×/2 < 3

D. 18 < x + 8

You cannot use the Hypotenuse-Leg congruence theorem to prove the similarity of triangles JKM and LKM, as they are not right triangles.

The Hypotenuse-Leg congruence theorem states that if the hypotenuse and one leg of two right triangles have the same measure then these two triangles are said to be congruent to each other.

The triangles JKM and LKM are not right triangles, as they do not have an angle of 90º, hence the HL congruence theorem cannot be used to prove their similarity.

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When evaluating the likelihood of an outcome, the claim or hypothesis statement is assumed. The required answer is a hypothesis claim or statement.

It is possible to test one's assumptions regarding a population parameter through the statistical analysis known as hypothesis testing. The association between two statistical variables can then be estimated by this hypothesis.

We start by writing the claim mathematically and determining if it is the null or alternative hypothesis. The null hypothesis is the assertion that there is no change or no relationship between the two variables; otherwise, the alternative hypothesis is the assertion that tells that there is a change or relationship.

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To show that the improper integral g(x)dx is divergent, but the average value of g on the interval 4, infinity is finite, we can consider a function g(x) = 1/x, which is mostly used as an example for a divergent integral.

The improper integral of g(x) from 4 to infinity is given by:

[tex]\int\limits {g(x)} \, dx = \int\limits {(1/x)} \, dx[/tex]  from 4 to infinity

= lim (upper limit -> infinity) [ln|x|] from 4 to x

= [ln|x|] from 4 to infinity

= ln(infinity) - ln(4) = infinity - ln(4) = infinity

As the limit of the integral is infinity, the integral is divergent.

However, the average value of g on the interval [4, ∞) is given by:

average value = (1/∞ - 4) * ∫(1/x)dx from 4 to infinity

= (1/∞ - 4) * (infinity - ln(4))

= -4 + ln(4)

= ln(4) - 4

which is a finite value when the integral is divergent

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The required blueberries Ben has 6 5/12 pounds.

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
Let the amount of blueberries Ben has to be x,
According to the question,
Together, they have 9 2/3 pounds of blueberries.
x + 3 1/4 = 9 2/3
x = 9 2/3 - 3 1/4
x = 9 + 2/3 - 3 -1/4
x = 6 + 5/12
x = 6 5/12 pounds

Thus, the required blueberries Ben has 6 5/12 pounds.

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Answer:

9.5

Step-by-step explanation:

Distance (d) = [tex]\sqrt{(2-(-7.5)^2)+(-11-(-11)^2)}\\ \sqrt{(9.5)^2+(0)^2}\\ \sqrt{90.25} \\9.5[/tex]

The transformation applied is the one in option D, a vertical dilation of scale factor of 5, and a shift of 10 units up.

Here we start with the parent quadratic function:

f(x) = x²

And we have the transformed function:

h(x) = 5x² + 10

We can write this as:

A vertical dilation of a scale factor 5, which will give:

h(x) = 5*f(x)

And then a translation of 10 units upwards, which gives:

h(x) = 5*f(x) + 10

Replacing the function f(x) we will get:

h(x) = 5*x² + 10

Then the correct option is D.

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Center of the sphere would be (-4.5, 1, -6)  and radius be 6.086.

Write the equation of the sphere?

A sphere's general equation is (x - a)2 + (y - b)2 + (z - c) 2 = r2, where (a, b, c) represents the sphere's center, r represents the radius, and x, y, and z are the coordinates of points on the sphere's surface.

To complete the square, we need to add and subtract the square of half of the x coefficient and y coefficient, and the square of half of the z coefficient.

The equation will be : [tex](x + 9/2)^2 + (y - 1)^2 + (z + 6)^2 = (9/2)^2 + 1^2 + 6^2[/tex]

So the standard form of the equation of the sphere is,    

          [tex](x + 4.5)^2 + (y - 1)^2 + (z + 6)^2 = 37[/tex]

The center of the sphere is (x, y, z) = (-4.5, 1, -6) and the radius is √37 ≈ 6.086

Therefore, Center of the sphere would be (-4.5, 1, -6) and radius be 6.086.

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A population of bacteria growing exponentially can be modeled as P(t)= P0e kt, where is the time in hours and P0 is the initial population. if the population has a doubling time of 3 hours, the growth constant  is k = ln(2) / 3 = 0.231049.

The growth constant (k) can be calculated by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours). Therefore, the growth constant is 0.231049.  This equation models exponential growth, which is a type of growth where the rate of increase is proportional to the population size. In this case, the population doubles every 3 hours.

This exponential growth is modeled using the equation P(t)= P0e^kt, where P(t) is the population size at a given time, P0 is the initial population size, and k is the growth constant.  By finding the growth constant, we can predict the population size at any given time. The growth constant is found by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours).

This equation is used to find the amount of growth per unit time and is the same regardless of the population size. This equation can be used to predict the population size at any given time, allowing us to estimate the population size in the future.

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The required number of sheets cut from the stock are 9.

A mixed fraction is one that is represented by both its quotient and remainder. For example, 2 1/3 is a mixed fraction, where 2 is the quotient, 1 is the remainder. An amalgam of a whole number and a legal fraction is a mixed fraction.

We have,

We need sheets of length = 2 (3/4) in from 26 (1/8) in stock.

So, 2 (3/4) = 11/4 in

26 (1/8) = 209/8

Then,

Number of sheets =  209/8 / 11/4

Number of sheets = 836/88

Number of sheets = 9.5

So, 9 sheets can be cut.

Thus, required number of sheets are 9.

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The solution to the equation 10+4y=-4y+2 is -1 because the variable y represents the same value.

An equation is a formula in mathematics that expresses the equality of two expressions by connecting them with the equals sign =. In its most basic form, an equation is a mathematical statement that shows that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical equation that depicts the relationship between two expressions on opposite sides of the sign. It mostly consists of one variable and one equal to symbol. 2x - 4 = 2 is an example.

Here,

10+4y=-4y+2

8y=-8

y=-1

The solution of equation 10+4y=-4y+2 is -1 as the variable y represents the same value.

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Answer:

To find the result of 2.4 divided by -0.06, we need to use the following steps:

Step 1: Change the division problem into a multiplication problem by flipping the divisor and multiplying by its reciprocal.

-0.06 ÷ 2.4 = -0.06 x 1/(2.4)

Step 2: To find the reciprocal of 2.4, we need to divide 1 by 2.4

1/(2.4) = 0.416666666666667

Step 3: Multiply the original dividend (2.4) by the reciprocal of the divisor (0.416666666666667)

2.4 x 0.416666666666667 = -0.1

So, 2.4 divided by -0.06 is equal to -0.1

It's worth noting that -0.1 is a negative number which makes sense since the divisor is negative, hence the result is the opposite of the result when dividing by a positive number.

Python lines to get the output for the probability P(x<0.325) are -

a) import scipy.stats as

st print(st.norm.cdf(0.325, 0, 1))

Hence, option a is the correct answer.

The Python code -

        import scipy.stats as

        st print(st.norm.cdf(0.325, 0, 1))

This line of code imports the scipy.stats module as "st", and then uses the norm.cdf function to calculate the cumulative distribution function (CDF) of a normal distribution with mean 0 and standard deviation 1, evaluated at x=0.325.

The CDF gives the probability that a random variable from the normal distribution is less than or equal to a certain value, so this line of code is outputting the probability P(x<0.325).

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Part (a)

The two limit statements tell us that this an exponential decay function.

The curve goes up forever when heading to the left (negative infinity) as indicated by the notation [tex]\displaystyle \lim_{\text{x}\to-\infty}f(x) = \infty[/tex]

At the same time, the curve slowly approaches the horizontal asymptote y = -2, when moving to the right, because of this notation [tex]\displaystyle \lim_{\text{x}\to\infty}f(x) = -2[/tex]

An exponential decay function like [tex]\text{y} = (0.5)^{\text{x}}[/tex] has a horizontal asymptote of y = 0, aka the x axis. The y value approaches 0 but never gets there. Each output is positive.

Shift everything down 2 units to arrive at [tex]\text{y} = (0.5)^{\text{x}}-2[/tex] and this will move the horizontal asymptote down the same amount.

There's nothing really special about the 0.5; you can replace it with any value in the interval 0 < b < 1.

---------

====================================================

Part (b)

I'll use this template

[tex]\text{y} = ab^{\text{x}}+c[/tex]

Plugging in x = 0 leads to y = a+c which is the y intercept. Set this equal to the stated y intercept 7 and we get a+c = 7.

We want the [tex]ab^{\text{x}}[/tex] portion to approach zero, which leads to c = 4 so we approach the stated horizontal asymptote.

So,

a+c = 7

a+4 = 7

a = 7-4

a = 3

We go from this

[tex]\text{y} = ab^{\text{x}}+c[/tex]

to this

[tex]\text{y} = 3b^{\text{x}}+4[/tex]

The value of b doesn't matter.

I'll go for b = 0.7 so we get to [tex]\text{f(x)} = 3(0.7)^{\text{x}}+4[/tex]

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====================================================

Part (c)

The parent function [tex]\text{y} = \log(\text{x}})[/tex] has a domain of [tex](0, \infty)[/tex]. In other words it is the interval [tex]0 < \text{x} < \infty[/tex]

If we replaced each input x with x-5, then we shift the xy axis 5 units to the left. It gives the illusion the log curve moves 5 units to the right.

The vertical asymptote also moves 5 units to the right. We go from a domain of [tex](0, \infty)[/tex] to a domain of [tex](5, \infty)[/tex]

The base of the log doesn't matter.

---------

Check out the graphs below. I used GeoGebra, but Desmos is another good option.

f(x) = 7x - 1, which is a linear equation with a positive leading coefficient, making f(x) > 0 for all x.

Set 1: {f(x) = x + 1, f(x) = x - 1}

Function f(x) = x^2 + 2 can be found in the span of the two given functions. This can be seen by expanding the equation:

f(x) = x^2 + 2 = (x + 1) + (x - 1) = 2x;

thus, f(x) = 2x + 2, which is a quadratic equation with a positive leading coefficient, making f(x) > 0 for all x.

Set 2: {f(x) = 4x - 2, f(x) = 3x + 1}

Function f(x) = 7x - 1 can be found in the span of the two given functions. This can be seen by expanding the equation:

f(x) = 7x - 1 = (4x - 2) + (3x + 1) = 7x;

thus, f(x) = 7x - 1, which is a linear equation with a positive leading coefficient, making f(x) > 0 for all x.

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Complete question:Find a function f(x) in the span of the functions f1(x) = x and f2(x) = -x such that f(x) > 0 for all x.

Answer:

  5.2 square units

Step-by-step explanation:

You want the area of a triangle with sides 4 and 3, and the angle between them measuring 60°.

The formula for the area of a triangle is ...

  A = 1/2ab·sin(C)

Here, we have a=4, b=3, C=60°, so the area is ...

  A = 1/2·4·3·sin(60°) = 3√3 ≈ 5.2 . . . . square units

The area of the triangle is about 5.2 square units.

The equation y is 2x - 3 describes the line with a slope of 2 and a Y-intercept of -3.

Use m is used to show the slope of a line.

m = 2

c is used to signify the line's y-intercept. c = -3

For the line's equation, the slope-intercept is: y = mx + c

In the equation above, replace c = -3 with m = 2 ,y = 2x - 3

Consequently, the equation of the line with a Y-intercept of -3 and a slope of 2 is as follows: y = 2x - 3.

For the line's equation, the slope-intercept is: y = mx + c

In the equation above, replace c = -3 with m = 2 ,y = 2x - 3

Consequently, the equation of the line with a Y-intercept of -3 and a slope of 2 is as follows: y = 2x - 3.

The complete question is,

The line with a -3 Y-intercept and a 2 slope has what equation?

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The amount each boy will have to contribute is $56.23

How to calculate the amount that each boy will contribute?

Andrew and his two brother want to contribute money together to buy a new game system

The cost of the game system is $168.69

Each boy want to contribute the same amount

Therefore the amount that each boy will contribute can be calculated as follows

There are 3 boys in total, Andrew and his two brothers

= 168.69/3

= 56.23

Hence each boy will contribute an amount of $56.23

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Answer:

less than or equal to N7500

Step-by-step explanation:

N2x - N5000 =  N2500

N2x = N7500

The solution of the given equation is X = B(C - A)⁻¹.

To solve for X in the equation AX + B = CX, we can rearrange the equation as follows:

AX - CX = -B

Factor out X on the left side:

(X(A - C)) = -B

To isolate X, we can multiply both sides of the equation by the inverse of (A - C), assuming it exists:

X = -(A - C)⁻¹ × B

X = (C - A)⁻¹ × B

Therefore, the solution for X is:

X = B(C - A)⁻¹

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The question 4 groups of 3 can be calculated to be a total of 12 objects across all groups.

The total number of groups are 4 in number

Each of the 4 groups have 3 objects in it.

That is 3 in 4 places

This can be solved by 4 * 3 = 12

Hence we can say that the solution for the questions that says 4 groups of 3 is 12

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The relative order of the metric prefixes Milli- < Centi- < Micro- < Kilo-

The relative order of the metric prefixes of kilo-, micro-, centi-, and milli- is milli-, centi-, micro-, and kilo-. This order is determined by the exponential values of each prefix.

Milli- is equal to 10-3, centi- is equal to 10-2, micro- is equal to 10-6 and kilo- is equal to 103. The exponential values are used to determine the relative order of the prefixes. The lowest exponential value is milli-, making it the smallest metric prefix and the highest exponential value is kilo-, making it the largest metric prefix.

When the exponential values are compared, it is clear that the order of the prefixes is milli- < centi- < micro- < kilo-. This order is used to determine the relative size of the metric prefixes when measuring the same base unit.

For example, if the base unit is meter, then millimeter is the smallest measure and kilometer is the largest measure. The exponential values of the metric prefixes determine the relative order of the base unit.

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The image of quadratic function f(x) = x² is equal to g(x) = 6 · x².

In this question we see the representation of quadratic equation f(x) = x², from which we need to generate the image of this function, this can be done by a rigid transformation known as vertical dilation:

g(x) = k · f(x), k > 1

Where:

If we know that f(x) = x² and k = 6, then the image of function is:

g(x) = 6 · x²

Whose representation on Cartesian plane is shown in the image attached below.

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Simple equation when u break it down: 33 x 24=792 apartments

A building, or edifice, is an enclosed structure with a roof and walls that is standing in one location more or less permanently, such as a house or factory (although there are also portable buildings). Buildings come in a variety of sizes, shapes, and uses, and they have been modified throughout history for a wide range of reasons, including the availability of building materials, weather, land prices, ground conditions, particular uses, prestige, and aesthetic considerations.

given:

new apartments' building =33 floor

apartments on each =24 floor

33*24=792

792 apartments are in the building

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Using Moore's Law, which states that the number of transistors on a chip doubles approximately every two years, the estimated number of transistors per IC in 1992 would be 64,000.

The law claims that we can expect the speed and capability of our computers to increase every two years because of this, yet we will pay less for them. In more simple terms, the observation by Gordon Moore in 1965 that the number of transistors in a dense integrated circuit (IC) doubles roughly every two years is known as Moore's law.

This is calculated by doubling the estimate of 4,000 transistors every two years for a total of 8 doublings (16 years).

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Answer:

x = 12

4 races

Step-by-step explanation:

2/3 = 8/x

Cross multiply.

2x = 8 × 3

x = 4 × 3

x = 12

Answer: x = 12

0.75 × 45 = 33.75

Since the number of races must be an integer, they must win a total of 34 races to have at least a 75% success rate.

They already won 30 races.

34 - 30 = 4

Answer: 4 races

The volume of the cylinder is 3,714.62 cubic yards.

We know that the volume of a cylinder of radius R and height H is given by:

V = pi*R²*H

where pi = 3.14

In this case, we also know that:

H = 7yd

And the diameter is 26 yards, the radius is half of that, then:

R = 26yd/2 = 13yd

Then the volume is:

V = 3.14*(13 yd)²*7yd = 3,714.62 yd³

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Quadrilateral DEFG is similar to quadrilateral HIJK The measure of side JK is 8.

Since quadrilateral DEFG is similar to quadrilateral HIJK, we know that the ratio of corresponding side lengths is equal. This means that if we know the length of one side of one of the quadrilaterals, we can use that ratio to find the length of the corresponding side in the other quadrilateral.

For example, if we know that side DE is 8 units long, we can use the ratio of side JK to side DE (x/8) to find the length of side JK. We can also use the same ratio of side IJ to side DF which is x/4.

Let's assume that the length of side DE is 8 units. We can use the ratio of side JK to side DE (x/8) to find the length of side JK.

[tex]x/8 = x/4[/tex]

Cross-multiplying and simplifying the equation we get

[tex]4x = 8x[/tex]

[tex]x = 8[/tex]

Therefore, the measure of side JK is 8 units.

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120 table spoons of lemon juice.

15 cups of olive oil.

Answer:

The equation for the area of a rectangle is length*width. For this problem, it says the length is 8 and the width is (3n+2). All you have to do is multiply 8 by (3n+2).

8(3n+2)=A

24n+16=A

Option A is correct