prove that every non-empty finite set has the same number of subsets of even size as it has subsets of odd size.

The quadratic equation is [tex]x^2+x-12[/tex]

What is quadratic equation?

Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations. The general form of the quadratic equation is:

ax² + bx + c = 0

where x is an unknown variable and a, b, c are numerical coefficients.

Given, the roots of quadratic equation as (3,-4).

The formula to find the quadratic equation, as

x2 – (Sum of the roots)x + Product of the roots = 0

Sum of roots = 3-4

=-1

Product of roots = 3(-4) = -12

The quadratic equation is

x²+x-12

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According to the given synthetic table, the quotient is 5x³ - 5x² + 3x + 2 and remainder is 1.

Synthetic table:

Basically, the shortcut for polynomial division when the divisor is of the form x – a. And only numeric coefficients of the dividend are used when dividing with synthetic division.

Given,

Here we have the synthetic table.

Now, we have to solve this one.

In order to find the quotient and remainder, we have to do the following steps:

Step-1 : Write down the first coefficient 5

Step-2 : Multiply our root -3 by our last result 5 to get -15 [ (-3) × 5=-15 ]

Step-3 : Add new result -15 to the next coefficient of the dividend 10, and write down the sum -5, [ 10 + (-15)=-5 ]

Step-4 : Multiply our root -3 by our last result -5 to get 15 [ (-3) × (-5)=15 ]

Step-5 : Add new result 15 to the next coefficient of the dividend -12, and write down the sum 3, [ (-12) + 15=3 ]

Step-6 : Multiply our root -3 by our last result 3 to get -9 [ (-3) × 3=-9 ]

Step-7 : Add new result -9 to the next coefficient of the dividend 11, and write down the sum 2, [ 11 + (-9)=2 ]

Step-8 : Multiply our root -3 by our last result 2 to get -6 [ (-3) × 2=-6 ]

Step-9 : Add new result -6 to the next coefficient of the dividend 7, and write down the sum 1, [ 7 + (-6)=1 ]

We have completed the table and have obtained the following coefficients 5,-5,3,2,1

Thus quotient is 5x³ - 5x² + 3x + 2  and remainder is 1.

And all these process are illustrated below.

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Answer: circumcenter

Step-by-step explanation:

If you meant the point of intersection of the perpendicular bisectors, this should be your answer.

Answer:

[tex]\textsf{D)} \quad y =-\dfrac{1}{3}x+7[/tex]

Step-by-step explanation:

The line of best fit of a scatter plot is a straight line that represents the general trend or pattern. It should pass as close as possible to the majority of the data points, minimizing the overall distance between the line and the points.

From observation of the given scatter plot, the line shows a negative association. Therefore, the slope of the line is negative. This means we can immediately discount the equations with positive slopes.

If we plot the line with the slope -2/3 (attachment 1 - blue line), we can see that the line passes through the first point, then has 3 points above the line but only 1 point below it.  The distance between point (6, 7) and the line is much greater than the distance between the line and the other points.

If we plot the line with the slope -1/3 (attachment 2 - red line), we can see that the line passes as close as possible to the majority of the data points, and minimizes the overall distance between the line and the points.

Therefore, the equation that is the best model for the line of best fit for the data is:

[tex]y =-\dfrac{1}{3}x+7[/tex]

The solution to equation (x+3)² - 5 = 0 is x = -3 +√5 and x = -3 - √5. The solution to equation (2x+1)(x-5)=0 is x = -1/2 and x = 5.

The quadratic function is defined as a function containing the highest power of a variable is two.

To determine the solution to equation (x+3)² - 5 = 0

(x+3)² - 5 = 0

x² + 9 + 6x - 5= 0

x²+  6x + 4 = 0

Using the quadratic formula to solve the above quadratic function

[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

[tex]x=\dfrac{-6\pm \sqrt{6^2-4\cdot \:1\cdot \:4}}{2\cdot \:1}[/tex]

[tex]\sqrt{6^2-4\cdot \:1\cdot \:4}[/tex] = 2√5

x = -6± 2√5 / 2

x = -3± √5

x = -3 +√5 and x = -3 - √5

To determine the solution to equation (2x+1)(x-5)=0

(2x+1)(x-5)=0

(2x+1) = 0 and (x-5)=0

2x = -1 and x = 5

x = -1/2 and x = 5

Thus, the solution to equation (x+3)² - 5 = 0 is x = -3 +√5 and x = -3 - √5. The solution to equation (2x+1)(x-5)=0 is x = -1/2 and x = 5.

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Option 3 is the solution. The direct variation in this question is gotten from the option where we have the input x: 4 6 8, output y: 6 9 12

In mathematics, the direct variation in the form that is

y = kx

To define the term k, k is the constant of proportionality.

The equation is written and interpreted as y is directly to x. At option 3,

y = kx

when y = 6 and x = 4, we would have 6 = 4k.

then k = 6/4

= 1.5

To confirm, we have to put the input values to see if they correspond with their y values.

1.5 x 6 = 9

1.5 x 8 = 12

Hence option 3.

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There is no difference between the mean IQ levels of people who were reared by their natural parents and those who were not.

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Answer:

-x^2+2x-8

Step-by-step explanation:

6x^2+2x-8-7x^2 Different features used to show like terms

Simplify =

-x^2+2x-8

Variance of the given distribution is equal to 0.63.

As given in the question,

Given data for the distribution is equal to :

X:            0          1           2            3

P(x):   0.027   0.189    0.441     0.343

Variance for the given distribution is equal to

= E(x²) - {E(x)}²

Now , Value of E(x) can be calculated using the formula,

E(x) = [tex]\sum x_{i} P( x_{i} )[/tex]

      = (0)(0.027) + (1)(0.189) + (2)(0.441) + (3)(0.343)

      = 0.189 + 0.882 + 1.029

      = 2.1

E(x²) = [tex]\sum x_{i} ^{2} P(x_{i})[/tex]

        =  (0)²(0.027) + (1)²(0.189) + (2)²(0.441) + (3)²(0.343)

        = 0.189 + 1.764 + 3.087

        = 5.04

Variance = E(x²) - {E(x)}²

               = 5.04 - (2.1)²

               = 5.04 - 4.41

               = 0.63

Therefore, variance of the given distribution is equal to 0.63.

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Step-by-step explanation:

6-(2x+4) is less than or equal to 10

6-(2x+4) ≤ 10

6 - 2x - 4 ≤ 10

combine like terms on left side:

2 - 2x  ≤ 10

subtract 2 from both sides:

2 - 2x - 2 ≤ 10 - 2

- 2x  ≤ 8

divide both sides by -2: (remember, when dividing by a negative number you must FLIP the sign direction)

- 2x/-2 ≥ 8/-2

x ≥ 8/-2

x ≥ -4

[-4, ∞)

Step-by-step explanation:

6-(2x+4) is less than or equal to 10

6-(2x+4) ≤ 10

6 - 2x - 4 ≤ 10

combine like terms on left side:

2 - 2x  ≤ 10

subtract 2 from both sides:

2 - 2x - 2 ≤ 10 - 2

- 2x  ≤ 8

divide both sides by -2: (remember, when dividing by a negative number you must FLIP the sign direction)

- 2x/-2 ≥ 8/-2

x ≥ 8/-2

x ≥ -4

[-4, ∞)

the circle's area is approximately 2173.22 mm².We can find the radius of the circle by dividing the diameter by 2

what is  diameter ?

Diameter is a straight line passing through the center of a circle or sphere and connecting two points on the circumference. It is the longest chord of the circle.

In the given question,

The diameter of a circle is 52.6 millimeters.

We can find the radius of the circle by dividing the diameter by 2:

radius = diameter/2 = 52.6/2 = 26.3 mm

Now, we can use the formula for the area of a circle:

area = π × radius²

Plugging in the values we have:

area = 3.14 × 26.3² = 3.14 × 691.69 ≈ 2173.21 mm²

Rounding to the nearest hundredth, we get:

area ≈ 2173.21 ≈ 2173.22 mm²

Therefore, the circle's area is approximately 2173.22 mm².

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A model to show [tex]\frac{9}{8}[/tex] is given below.

What is an improper fraction?

When the numerator of a fraction is more than or equal to the denominator, the fraction is said to be improper. The number at the top is the numerator, and the number at the bottom is the denominator.

Given, improper fraction = [tex]\frac{9}{8}[/tex]

Now, convert the given improper fraction into a mixed fraction

i.e.  [tex]\frac{9}{8} = 1\frac{1}{8}[/tex] ( mixed fraction)

That we can write [tex]1\frac{1}{8} = 1 + \frac{1}{8}[/tex]

The model to show [tex]\frac{9}{8}[/tex] is given below:

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The probability that Alice leaves before Carol is 0.5 according to the given information.

Probability is a branch of mathematics that deals with numerical descriptions of how likely an event or statement is to occur.

If X1 and X2 are both exponential distributions with parameters 1 and 2, respectively, then

P(X₁< X₂)= λ₁/(λ₁+λ₂ )

And P(X>s+t/x>s)=P(X>t), if X≈exp(μ)

Here, Let [tex]X_{A}[/tex], [tex]X_{C}[/tex] denote the service times for Alice and Carol

And also given that,

E[ [tex]X_{A}[/tex]]=1/μ

E[ [tex]X_{A}[/tex]]=4

E[ [tex]X_{C}[/tex]]=4

E[ [tex]X_{A}[/tex]]=E[ [tex]X_{C}[/tex]]=4

So, [tex]X_{A}[/tex], [tex]X_{C}[/tex] ≈exp(1/4)

Now, when Betty completes his service, Alice is still being served, and the remaining time required to complete his service is exp(1/4).

Because the exponential distribution has no memory.

So, P( [tex]X_{A}[/tex]< [tex]X_{C}[/tex] )=  λ₁/ λ₁+ λ₂

=(1/4)/((1/4)+(1/4))

=(1/4)/(1/2)

=0.5

As a result, the probability that Alice leaves before Carol is 0.5.

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Answer:

4/5

Step-by-step explanation:

hope it helped :)

Answer:

x ≈ 28.3

Step-by-step explanation:

using the cosine ratio in the right triangle

cos45°= [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{x}{40}[/tex] ( multiply both sides by 40 )

40 × cos45° = x , then

x ≈ 28.3 ( to the nearest tenth )

There are five rules in one group

From the question, we have

25 rules in 5 groups

In one group = 25/5 = 5 rules

There are five rules in one group

Divide:

Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. When dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers. One of the four fundamental mathematical operations, along with addition, subtraction, and multiplication, is division. Division is the process of dividing a larger group into smaller groups so that each group contains an equal number of things. It is a mathematical operation used for equal distribution and equal grouping.

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A class is made up of 6 boys and 14 girls. half of the boys wear glasses. a student is selected at random from the class.

0.2% probability that the student is a boy with glasses.

In order to calculate a probability we first need to know the total possible outcomes, so the total is boys plus the girls, 6+14=20, we have a total of 20 students, and we need the total number of successful outcomes, which is half of the girls, 4 girls use glasses.

So the probability is:

[tex]Probability = \frac{Successful Outcomes}{Total Possible outcomes}[/tex]

[tex]Probability = \frac{4}{20} \\\\Probability = \frac{1}{5} \\\\Probability = 0.2[/tex]

So the probability is 0.2% that if you pick a student at random you will pick a girl with glasses.

Hence the answer is 0.2% probability that the student is a boy with glasses.

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The percent of times that Emerson hit the ball, given the number of times that he struck out was 68 %

First, find the total number of times that Emerson had at bat:

= Game 1 + Game 2 + Games 3

= 90 + 120 + 140

= 350 times

The number of times Emerson missed was:

= 30 + 40 + 42

= 112 times

The number of times the ball was hit is therefore:

= 350 - 112

= 238 times

The percent of time the ball was hit:

= 238 / 350

= 68 %

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On simplifying the given equations y = (x - 2)² - 9 and y = (x + 1)(x - 5), they  represent the same parabola which is y = x² - 4x - 5

What is equation of parabola?

The equation of a parabola can be expressed in standard form and vertex form. The standard form of a parabola is y = ax² + bx + c where a, b, and c are constants, and a does not equal 0.

The vertex form of a parabola is y = a(x - h)² + k. The coordinates (h,k) represent the vertex of the parabola.

We can check if the given equations represent the same parabola by simplifying them to standard form of equation of parabola.

According to the given equation:

y = (x - 2)² - 9

y = x² - 4x + 4 - 9

∴ y = x² - 4x - 5 ------ 1)

Similarly simplifying other equation

y = (x + 1)(x - 5)

y = x² - 5x + x - 5

y = x² - 4x - 5 ----- 2)

Clearly equation 1) and 2) represent the same parabola having equation

y = x² - 4x - 5

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theorem of sines answer: A

As we have divided the denominator by 5, we must also divide the numerator by 5. 10 dog treats divided by 5 beagles gives us 2 dog treats per Beagle. So our Unit Rate is Unit Rate of Dog Treats per beagle is = 2:1 2:1 is a direct ratio of 10:5, they are in fact equivalent ratios.

Answer:

2:1

Step-by-step explanation:

10/5 is equal to 2/1 simplified. Therefore the simplified ratio of 10/5 is 2:1

The amount owed is $10865, based on the given parameters

The given parameters about the compound interest are

Principal Amount, P = $5300

Interest Rate, R = 8%

Time, t = 6 years

To calculate the amount owed, we make use of the following formula

A = P + CI

Where

Compound interest, CI = P(1 + R)^t - P

So, the equation becomes

A = P + P(1 + R)^t - P

Evaluate the like terms

A = P(1 + R)^t

In this question, the formula is given as

A = P(1 + r/n)^nt

Where

n = 12, i.e compounded monthly

Substitute the known values in the above equation

A = 5300 * (1 + 8%/12)^(6*12)

Express 8% as decimal

A = 5300 * (1 + 0.08/12)^(6*12)

Evaluate the sum

A = 5300 * (1.01)^(6*12)

Evaluate the exponent

A = 5300 * 2.05

Evaluate the product

A = 10865

Hence, the value of the amount after 6 years is $10865

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In order to solve for the variable in the equation

We want to apply the distributive property to the given expression.

1 - x - 2 + 2x = 10x - 25 - x is the result of this step.

Let's see how to solve this:

Remember that the distributive property says that:

A*(B + C) = A*B + A*C.

Here we have the expression:

1 - (x + 2) + 2x = 5*(2x - 5) - x

We can apply the distributive property to the first term in the right side, we will get:

1 - (x + 2) + 2x = 5*(2x - 5) - x

1 - (x + 2) + 2x = 5*2x - 5*5 - x

1 - (x + 2) + 2x = 10x - 25 - x

Now we can also apply the distribute property to the second term in the left side:

1 - (x + 2) + 2x = 10x - 25 - x

1 - x - 2 + 2x = 10x - 25 - x

Hence the answer is, 1 - x - 2 + 2x = 10x - 25 - x is the result of this step.

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Answer:

  (a) (x, y) = (2, 5)

  (bi) y = 5, y = 5; the values are the same

  (bii) the values are the same as in part (a)

Step-by-step explanation:

You want the solution and some observations about it for the system of equations ...

The graph shows the lines cross at the point (x, y) = (2, 5).

The solution to the equations is (x, y) = (2, 5).

i) For x=2, the value of y in the first equation is ...

  y = 2(2) +1 = 4 +1 = 5

For x=2, the value of y in the second equation is ...

  y = -(2) +7 = -2 +7 = 5

ii) The values of y are the same, and match the value we found in part (a).

__

Additional comment

The purpose of the exercise is to get you to see that the value of x that is the solution makes both equations have the same value of y, and that is the value of y that is the solution.

The graph is a plot of the y-value for each x-value. Where the lines cross, the y-values are the same for that x-value. That point of intersection of the lines is the solution to the system of equations: the values of x and y satisfy both equations at that point.

Answer: y=$4.50+$3.20x

Step-by-step explanation: You already have to pay $4.50 plus $3.20 every mile driven (x) equals your final cost (y)

If reserve rate decreases from 5% to 4%  then money supply of $50000

deposits increases .i.e bank can save $2000 and can lend $48000.

The reserve ratio is the percentage of reserves (or cash deposits) that a bank must keep and not lend out. The higher the reserve requirement, the less money a bank can potentially lend—but this extra cash also prevents a bank from failing and strengthens its balance sheet. Still, when the reserve ratio rises, monetary policy is considered contractionary, and when it falls, monetary policy is considered expansionary.

Given this, the reserve rate falls from 5% to 4%.

$50000 is the initial deposit.

If the reserve ratio is 5%, the bank must keep 50000*0.05 = $2500 in reserve and may make loans totaling 50000-2500=$47500.

If the reserve ratio is reduced to 4%, a bank with $50000 in deposits must save 50000*0.04 = $2000 before it can lend $48000.

Thus, if reserve ratio decreases then the money supply increases.

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Yes, the electron pair repulsion determines the molecular shape, we can use  VSEPR to find out how.

What is VSEPR theory?

The amount of valence electron pairs present in the outermost shell is used by the valence shell electron repulsion theory (VSEPR) to predict the structure of molecules.

The number of valence shell electron bond pairs between atoms in a molecule is used in the valence shell electron repulsion theory (VSEPR), a model for predicting 3D molecular form.

VSEPR theory predicts that BeF2 should be a linear molecule, with a 180o angle between the two Be-F bonds.

The VSEPR theory, therefore, predicts a trigonal planar geometry for the BF3 molecule, with an F-B-F bond angle of 120o.

we can predict that a tetrahedral molecule in which the H-C-H bond angle is 109o28'.

Hence we can predict the molecular shape, using VSEPR theory.

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Answer:

See below

Step-by-step explanation:

Answer:

slope = - [tex]\frac{5}{4}[/tex]

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (6, - 3 ) and (x₂, y₂ ) = (- 2, 7 )

m = [tex]\frac{7-(-3)}{-2-6}[/tex] = [tex]\frac{7+3}{-8}[/tex] = [tex]\frac{10}{-8}[/tex] = - [tex]\frac{5}{4}[/tex]

The slope of a line that passes through the points (6,-3),(-2,7) is -5/4.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope of a line that passes through the points (6,-3),(-2,7):

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

m = (7+3)/(-2-6)

m = -10/8

m = -5/4

Therefore, the slope is -5/4.

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