Quick algebra 1 question for 25 points!

Only answer if you know the answer, Tysm!

Quick Algebra 1 Question For 25 Points!Only Answer If You Know The Answer, Tysm!

Answers

Answer 1

Using a quadratic regression equation, it is found that the prediction of y when x = 6 is of -154.70.

How to find the equation of quadratic regression using a calculator?

To find the equation, we need to insert the points (x,y) in the calculator.

In this problem, we have that the function initially increases, then it decreases, which means that it is a quadratic function. The points (x,y) to  be inserted into the calculator are given as follows:

(-1, -2.75), (0, 2.15), (1, -1.60), (2, -13.85), (3, -34.55).

Inserting these points into the calculator, the prediction of y for a value of x is given as follows:

y = -4.721428571x² + 1.482857143x + 3.201428571

Hence, when x = 6, the prediction is:

y = -4.721428571 x 6² + 1.482857143 x 6 + 3.201428571

y = -154.70

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Related Questions

The formula for __________________________ is x = t e or raw score (x) equals the true score (t) plus error (e).

Answers

The formula for Observed Score is; Observed score (X) = True score (T) + measurement error (e)

What is the observed score formula?

The formula for Observed Score is;

Observed score (X) = True score (T) + measurement error (e)

1) True score is the score that would be obtained if an individual took a test an infinite amount of times and those test scores were averaged. The concept of true score is theoretical because you can't give someone something an infinite number of times.

2) Standard error of measurement is the standard deviation of multiple test scores (how far the sample mean of the data is likely to be from the true population mean).

The less random error (e) in the measure, the more the observed score X approximates the true score T.

Thus;

Observed score = True Score + Error

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The data set below has a lower quartile of 13 and an upper quartile of 37.

1, 12, 13, 15, 18, 20, 35, 37, 40, 78

Which statement is true about any outliers of the data set?

Answers

The correct option regarding the outliers of the data-set is given by:

The greatest value, 78, is the only outlier.

How to use the quartiles of a data-set to identitfy outliers?

The median of the data-set separates the bottom half from the upper half, that is, it is the 50th percentile.The first quartile is the median of the first half of the data-set.The third quartile is the median of the second half of the data-set.The interquartile range is the difference of the third quartile with the first quartile.Measures that are more than 1.5 IQR from Q1 and Q3 are considered outliers.

The IQR for this problem is:

IQR = 37 - 13 = 24.

Hence the bounds for outliers are:

Less than 13 - 1.5 x 24 = -23.Greater than 37 + 1.5 x 24 = 73,

The options are:

No outliers.Only 1 is an outlier.Only 78 is an outlier.Both 1 and 78 are outliers.

Hence the correct option is that only 78 is an outlier.

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Look at the picture of a scaffold used to support construction workers. The height of the scaffold can be changed by adjusting two slanting rods, one of which, labeled PR, is shown:

A support structure is shown in which a right triangle PQR is formed with the right angle at Q. The length of PQ is shown as 14 feet, and the length of QR is shown as 6 feet..

Part A: What is the approximate length of rod PR? Round your answer to the nearest hundredth. Explain how you found your answer, stating the theorem you used. Show all your work. (5 points)

Part B: The length of rod PR is adjusted to 16 feet. If width PQ remains the same, what is the approximate new height QR of the scaffold? Round your answer to the nearest hundredth. Show all your work. (5 points)

Answers

Part A

Using the Pythagorean on the right triangle PQR, with PQ and QR as the legs and PR as the hypotenuse,

[tex]14^2 +6^2 =(PR)^2\\\\(PR)=\sqrt{14^2 +6^2}\\\\PR \approx \boxed{15.23 \text{ ft}}[/tex]

Part B

[tex](QR)^2 +6^2 =16^2\\\\(QR)^2 =16^2 -6^2\\\\QR=\sqrt{16^2 -6^2}\\\\QR \approx \boxed{14.83 \text{ ft}}[/tex]

Evaluate the following series:

Answers

This is a telescoping sum. The K-th partial sum is

[tex]S_K = \displaystyle \sum_{k=1}^K \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) \\\\ ~~~= \left(\frac1{\sqrt2} - \frac1{\sqrt4}\right) + \left(\frac1{\sqrt3} - \frac1{\sqrt5}\right) + \left(\frac1{\sqrt4} - \frac1{\sqrt6}\right) + \left(\frac1{\sqrt5} - \frac1{\sqrt7}\right) + \cdots \\\\ ~~~~~~~~+ \left(\frac1{\sqrt{K-1}} - \frac1{\sqrt{K+1}}\right) \\\\ ~~~~~~~~+ \left(\frac1{\sqrt K} - \frac1{\sqrt{K+2}}\right) + \left(\frac1{\sqrt{K+1}} - \frac1{\sqrt{K+3}}\right)[/tex]

[tex]\displaystyle = \frac1{\sqrt2} + \frac1{\sqrt3} - \frac1{\sqrt{K+2}} - \frac1{\sqrt{K+3}}[/tex]

As [tex]K\to\infty[/tex], the two trailing terms will converge to 0, and the overall infinite sum will converge to

[tex]\displaystyle \sum_{k=1}^\infty \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) = \lim_{k\to\infty} S_k = \boxed{\frac1{\sqrt2} + \frac1{\sqrt3}}[/tex]

By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] -  [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.

Is the series convergent?

Herein we have a series that involves radical components. First, we simplify the expression given:

∑ [1 / √(k + 1) - 1 / √(k + 3)] = ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)]

The convergence of the series can be proved by the limit comparison test, where each component of the subtraction of the series is compared with a series that is convergent. We notice that both 1 / √(k + 1) and 1 / √(k + 3) resembles the expresion 1 /√k. Then, we have the following subtraction of ratios:

[1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k]

√k / √(k + 1) - √k / √(k + 3)

√[k / (k + 1)] - √[k / (k + 3)]

Then, by using the limit property for rational functions we find the following result for n → + ∞:

√[1 / (1 + 0)] - √[1 / (1 + 0)]

√1 - √1

1 - 1

0

By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] -  [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.

Remark

The statement is incomplete and complete form cannot be found, therefore, we decided to determine if the series is convergent or not.

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pls help will mark this brainlest

Answers

The ordered pairs (0,13) and (10, 0) are joined to best draw the line of best fit for the given scatter plot. So, option 4 is correct.

How to draw the line of best fit for a scatter plot?

For the given scatter plot, to draw a line of best fit, the slope is to be calculated. The slope of the required line is calculated by

m = [n(∑xy) - (∑x)(∑y)]/[n(∑x²) - (∑x)²]

Where,

∑xy = sum of the product of x and y values

∑x = sum of x values

∑y = sum of y values

∑x² = sum of square values of x

n = total number of scatter points

And the y-intercept is calculated by

b = [∑y - m(∑x)]/n

Where m is the slope obtained above

Calculation:

The given scatter plot has the coordinate points:

(0,14), (1, 11), (2, 9), (3, 10),(4, 7), (5, 7), (6, 5), (7, 5), (8, 3), (9, 1), (10, 0)

Such that n = 11

Then the required components are calculated as follows:

∑x = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

∑y = 14 + 11 + 9 + 10 + 7 + 7 + 5 + 5 + 3 + 1 + 0 = 72

∑xy = (0 × 14) + (1 × 11) + (2 × 9) + (3 × 10) + (4 × 7) + (5 × 7) + (6 × 5) + (7 × 5) + (8 × 3) + (9 × 1) + (10 × 0) = 220

∑x² = 0² + 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385

Then the slope is calculated as follows:

slope m =  [n(∑xy) - (∑x)(∑y)]/[n(∑x²) - (∑x)²]

On substituting,

m = [11(220) - (55)(72)]/[11(385) - (55)²]

⇒ m = -14/11 = -1.2727273 ≅ -1.3

∴ m = -1.3

Then calculating the y-intercept:

we have b = [∑y - m(∑x)]/n

On substituting,

b = [72 - -1.3(55)]/11

∴ b = 13

Then the slope-intercept form of the required line is

y = -1.3x + 13

When x = 0,

y = -1.3(0) + 13 = 13

When y = 0,

0 = -1.3x + 13

⇒ 1.3x = 13

⇒ x = 13/1.3 = 10

Therefore, the coordinates (0, 13) and (10, 0) give the best draw for the line of best fit.

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Help on this math problem!! if there is any work that can be shown it would be great thanks!!!

Answers

I’m just assuming this but I think it’s 64.4 because I think all of the subtraction are taken away by 38 so I just took 102.4-38

The diameter of a circle graphed in the xy-plane has endpoints at (-2, -1) and (4,7). Which of the following is an equation of the circle?
A (x + 1)² + (y + 1)² = 25
B (x + 1)² + (y + 1)² = 100
C (x - 1)² + (y - 3)² = 100
D (x - 1)² + (y - 3)² = 25 ​

Answers

The required equation of the circle whose endpoint of the diameter  (-2, -1) and (4,7) is  (x - 1)² + (y - 3)² = 25 ​. Option D is correct.

Given that,

The endpoints of the circle's diameter (-2, -1) and (4,7).
The equation of the circle is to be determined.


What is a circle?

The circle is the locus of a point whose distance from a fixed point is constant i.e center (h, k). The equation of the circle is given by

(x - h)² + (y - k)² = r².

where h, k is the coordinate of the circle's center on the coordinate plane and r is the circle's radius.

Center of the circle(h, k) = (-2 + 4)/2 ,(-1+7)/2
                                  = 1  , 3
Radius of the circle,
[tex]r^2 =\sqrt{(1+2)^2+(3+1)^2}\\r^2 =\sqrt{9+16} \\r^2 = 25[/tex]


Equation of the circle with center (1, 3)
[tex]r^2 = (x - 1)^2 + (y-3)^2[/tex]
[tex]25 = (x - 1)^2 + (y-3)^2[/tex]

Thus, the required equation of the circle whose endpoint of the diameter  (-2, -1) and (4,7) is  (x - 1)² + (y - 3)² = 25 ​. Option D is correct.


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Express the volume of the box as a
polynomial in the variable x

Answers

The volume of the box as a polynomial in the variable x is x(12 - 2x)(7 - 2x)

How to determine the volume?

The complete question is added as an attachment

From the attached image, we have:

Length = 12 - 2x

Width = 7 - 2x

Height = x

The volume is calculated as:

Volume = Length * Width * Height

Substitute the known values in the above equation

Volume = (12 - 2x) * (7 - 2x) * x

This gives

Volume = x(12 - 2x)(7 - 2x)

Hence, the volume of the box as a polynomial in the variable x is x(12 - 2x)(7 - 2x)

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(MID SCHOOL ALGEBRA)
can somebody explain to me from this image, why the -13x and +6 becomes +13x and -6?

Answers

Answer:

- (2x^2 - 13x + 6)

The above expression is in brackets and a negative sign is outside of the brackets. This means that when expanding the expression, we have to multiply each number with the negative sign (each number would switch their signs to the opposite one). Therefore, 2x^2 becomes -2x^2, -13x becomes 13x and 6 becomes -6.

Basically it the same as multiplying by -1
- * - = + and - * + = -

which number set(s) does -10 belong to

irrational numbers
whole numbers
rational numbers
integers
real numbers
counting or natural numbers
No number set describes this number.

Answers

The number set(s) that - 10 belong to are rational numbers, integers and real numbers. Option C, D and E

Number sets of negative numbers

A rational number can be defined as a number expressed as the ratio of two integers, where the denominator is not be equal to zer0

- 10 can be written as = 1/ 10

Integers are whole number that could be positive, negative and even zero

- 10 is a negative whole number

Real numbers are numbers with continuous quantity that can represent  distance along a number line

-10 can represent distance along a number line.

Thus, the number set(s) that - 10 belong to are rational numbers, integers and real numbers. Option C, D and E

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Which of the following is a solution to the equation
3/4
x-12 = -18?


1) 4.5
2) -8
3) -22.5
4) -40

Answers

Answer: B) -8

Step-by-step explanation:

3/4 x -12 = -18

We add 12 both sides

3/4 x = -18 + 12

Add −18 and 12 to get −6.

3/4 x = -6

Multiply both sides by 3/4 by the reciprocal of 3/4.

x = -6 × (3/4)

Express a -6 x (3/4) as a single fraction.

x = -6 × 4 / 3

Multiply −6 and 4 to get −24.

x = -24 / 3

Divide −24 by 3 to get −8.

x = -8

Answer: Option "2" is correct. ✅

How should I solve this?

Answers

The parallel sides AB, PQ, and CD, gives similar triangles, ∆ABD ~ ∆PQD and ∆CDB ~ ∆PQB, from which we have;

[tex] \frac{1}{x} + \frac{1}{y}= \frac{1}{z}[/tex]

Which method can be used to prove the given relation?

From the given information, we have;

∆ABD ~ ∆PQD∆CDB ~ ∆PQB

According to the ratio of corresponding sides of similar triangles, we have;

[tex] \frac{x}{z} = \mathbf{\frac{BD}{QD} }[/tex]

[tex] \frac{y}{z} = \frac{BD}{ BQ} [/tex]

Which gives;

[tex] \mathbf{\frac{y}{z}} = \frac{BD }{ BD - Q D} [/tex]

[tex] \frac{z}{y} = \frac{BD - QD }{ BD } = 1 - \frac{Q D }{ BD}[/tex]

QD × x = BD × z

BD × z = (1 - QD/BD) × y = y - (QD × y/BD)

Therefore;

BD × z = y - (QD × y/BD)

BQ × y = y - (QD × y/BD)

BQ × y = y - (z × y/x) = y × (1 - z/x)

(1 - z/x) = BQ

BD × z = y × (1 - z/x)

BD = (y × (1 - z/x))/z

Therefore;

QD × x = y × (1 - z/x)

(BD-BQ) × x = y × (1 - z/x)

(BD-(1 - z/x)) × x = y × (1 - z/x)

BD = (y × (1 - z/x))/x + (1 - z/x)

BQ + QD = (1 - z/x) + (y × (1 - z/x))/x

BD = BQ + QD

(y × (1 - z/x))/x + (1 - z/x) = (y × (1 - z/x))/z

(1 - z/x)×(y/x + 1) =(1 - z/x) × y/z

Dividing both sides by (1 - z/x) gives;

y/x + 1 = y/z

Dividing all through by y gives;

(y/x + 1)/y = (y/z)/y

1/x + 1/y = 1/z

Therefore;

[tex] \frac{1}{x} + \frac{1}{y}= \frac{1}{z}[/tex]

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A company currently pays a dividend of $2.6 per share (d0 = $2.6). it is estimated that the company's dividend will grow at a rate of 24% per year for the next 2 years, and then at a constant rate of 8% thereafter. the company's stock has a beta of 1.8, the risk-free rate is 7.5%, and the market risk premium is 4.5%. what is your estimate of the stock's current price?

Answers

The current value of the stock is $53.413455.

What is the CAMP model?The CAMP model describes the relationship between systemic risk, or the general dangers of investing, and projected return on assets, specifically stocks.

To find the current price of stock:

Stock’s current price: 53.41 dollars

First, we determine the stock's value using the CAPM model.

Ke = rf + (rm - rf)risk free, rf =  0.085.premium market = (rm - rf) = 0.045(nondiversifiable risk) = 1.3

Then, Ke = 0.085 + 1.3(0.045).

Ke = 0.14350

The current value of the future dividends must now be known:

D0 = 2.8D1 = D0 x (1 + g) = 2.8 * 1.23 = 3.444D2 =  3.444 x 1.23 = 4.2361200

The next dividends, which are at perpetuity will be solved using the dividend growth model:

dividends/return-growth = Intrinsic value

In this case, dividends will be:

4.23612 x 1.07 = 4.5326484

Return will be calculated using the CAPM and g = 7%.

plug this into the Dividend grow model,

4.53264840.1435-0.07 = Intrinsic ValueValue of the dividends at perpetuity: 61.6686857

Finally, it's crucial to remember that these statistics were calculated for the present year.

We must bring them to the present day using the present value of a lump sum:

principal/(1 + rate)time = PV3.444/(1 + 0.1435)1 = PV3.011805859 = PV4.23612/(1 + 0.1435)2 = PV3.239633762 = PV61.6686857/(1 + 0.1435)2 = PV47.16201531 = PV

We add them and get the value of the stock, 53.413455

Therefore, the current value of the stock is $53.413455.

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3m
2. What is the volume of this object?
1m
2 m
6 m
4 m
3 m

Answers

Answer:

60m3

Step-by-step explanation:

my son got the right anser

Answer:

60m^3

Step-by-step explanation:

The formula for volume of a rectangular prism is height*width*depth

We can see two distinct shapes here

First, we find the volume for the bigger shape

We can see the measurements that pertain to it...

Width is 6m, depth is 3m, and height is also 3m

Multiply those three, you get 54m^3

Now the smaller shape...

1m in width, 2m in height, and 3m in depth

Multiply those, get 6m^3

Add the two shapes together, 54+6=60m^3

The quotient equals the divisor, then the dividend equals the
(A) √divisor
(B) divisor
(C) divisor²
(D) quotient

Answers

a / b = c
a is dividend
b is divisor
c is quotient

c = b then
a / b = b

a = b^2 or
divisor squared
divisor ^ 2

Type the correct answer in each box. Round your answers to two decimal places.
Subtract vector v = <2, -3> from vector u = <5, 2>.
The magnitude of the resulting vector, u - v, is approximately __
and its angle of direction is approximately ___

Answers

The magnitude of the resulting vector, u - v, is approximately 5.83

and its angle of direction is approximately 59.04°.

How to find the magnitude of the resulting vector?

We want to subtract vector v from vector u.

We are given;

v = <2, -3> = 2i - 3j

u = <5, 2> = 5i + 2j

u - v = 5i + 2j - (2i - 3j)

= 5i + 2j - 2i + 3j

= 3i + 5j

Resultant vector = √(3² + 5²)

Resultant vector = √34 ≈ 5.83

Angle of direction of resultant vector is;

tan θ = (5/3)

θ = tan⁻¹(5/3)

θ = 59.04°

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A scale measured a 4.5-pound brick as weighing 5.3 pounds. which measurement is more accurate but less precise than 5.3 pounds? 4.98 pounds 5 pounds 5.52 pounds 6 pounds

Answers

The correct option is B.

5 pound

The number (5.3) after Decimal is <5 so it round off to 5.

What is the property of rounding off numbers to one decimal place?

It is the same to round a number to one decimal place as to the nearest tenths. At this instance, it is known which numeral is in the hundredths position. When the number in the hundredths place is higher than or equal to 5, the tenths digit is raised by one unit.

How do you round off decimals examples?

A typical rule of thumb is to glance at the digit immediately to the right of the place value you want to round to and make your choice. For instance, rounding 5.1837 to the closest hundredth would result in 5.18 (because 3<5), whereas rounding 5.184 to the nearest thousandth would result in 5.184 (because 7>5).

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I understand that the question your are looking for is:

A scale measured a 4.5-pound brick as weighing 5.3 pounds. which measurement is more accurate but less precise than 5.3 pounds?

A. 4.98 pounds

B. 5 pounds

C. 5.52 pounds

D. 6 pounds

Consider a single spin of the spinner. A spinner contains 4 equal sections: 1, 2, 4 and 3. Sections 1 and 4 are shaded. The spinner is pointed at number 2. Which events are mutually exclusive

Answers

Answer:

Events that mutually exclusive are 1. Landing on a shaded portion and landing on a 3.2. Landing on an unshaded portion and landing on a number less than 2.

Choose the numbers that are terminating decimals. Select all that apply.
A. 0.032
B. 0.999...
C. 0.525
D. 0.75

Answers

Answer: A, C, and D

Step-by-step explanation:

          A terminating decimal is a decimal that has an end. In other words, [tex]\frac{1}{4} =0.25[/tex] is one, but [tex]\frac{1}{3} =0.3333...[/tex] is not.

✓ A. 0.032

✗ B. 0.999...

✓ C. 0.525

✓ D. 0.75

HELP HELP PLEASEEE

Theo has 2.5k followers. He knows that if he posts daily, he will gain 1k followers each day.
In this scenario, identify the initial value and rate of change. Then, explain why this is or is not
proportional and/or a linear relationship.

Answers

Bsbsnsnsjsjenend 2828282828282828289229292829

what is the area
please help me

Answers

The area and perimeter of the picture is 576 inches square and 96 inches respectively

How to determine the area

Given that each frame is a rectangle and four frames makes up the picture

Note that the picture takes the shape of a square

Let's find the total perimeter

Perimeter of the picture = sum of the four rectangular frame perimeters

Perimeter = 24 + 24 + 24 + 24

Perimeter = 96 inches

Formula for area of a square = a^2

Where 'a' is the length of the side  = 24 inches

Substitute the value of 'a'

Area of the picture = 24^2

Area of the picture = 576 inches square

Thus, the area and perimeter of the picture is 576 inches square and 96 inches respectively.

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Which element of expectancy theory could be phrased as the question, "what’s the probability that, if i do a good job, that there will be some kind of outcome in it for me?"

Answers

The expectancy theory that could be phrased as the above question is based on the element called expectancy.

In this question,

Expectancy theory proposes that an individual will behave or act in a certain way because they are motivated to select a specific behavior over others due to what they expect the result of that selected behavior will be.

To make the connection between motivation, effort and performance, expectancy theory has three variables: Expectancy, Instrumentality and Valence.

Expectancy theory consists of three basic components:

(1) The employee's expectancy that working hard will lead to his or her desired level of performance;

(2) The employee's expectancy that working hard will thus ensure that rewards will follow; and

(3) Whether or not the employee's perception that the outcome of working hard is worth the effort or value associated with hard work.

VIE expectancy theory is often formulated using the equation MF (motivational forces) = V (valence) x I (instrumentality) x E (expectancy).

From the definition, the probability that, "if i do a good job, that there will be some kind of outcome in it for me" is comes under the element expectancy(E).

Hence we can conclude that the expectancy theory that could be phrased as the above question is based on the element called expectancy.

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What number could be added to 0.40 ml for the level of precision to be 0.01 ml? check all that apply 0.2 ml 0.154 ml 6 ml 2.02 ml 8.8331 ml

Answers

The following volumes of fluid can be added to 0.40 mL of liquid for the level of precision of 0.01 mL: a) 0.2 mL, b) 6 mL, c) 2.02 mL. The amounts 0.154 mL and 8.8331 mL are not possible due to given level of precision.

What is the amount of liquid to be added to sample according to a given precision?

If the measuring has a level of precision of 0.01 ml, this means that the measured quantities are only sensible to the smallest hundreths. Any change less than 0.01 ml and any decimal less than a hundreths are "invisible" for measuring processes.

Hence, the following volumes of fluid can be added to 0.40 mL of liquid for the level of precision of 0.01 mL: a) 0.2 mL, b) 6 mL, c) 2.02 mL. The amounts 0.154 mL and 8.8331 mL are not possible due to given level of precision.

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Answer:

B,D,E

Step-by-step explanation:

Alex travels 46
miles per hour for
3.2 hours. How far
has he gone?

Answers

46 mph * 3.2 h = 147.2 miles

Answer: 147.2 miles

Step-by-step explanation:

We can answer this by knowing the formula [tex]speed=\frac{distance}{time}[/tex]. Here, the speed is 46 miles/hour and the time is 3.2 hours. Let's put these values into the formula and solve for distance.

[tex]46=\frac{distance}{3.2}\\46*3.2=\frac{distance}{3.2}*3.2\\d=147.2[/tex]

Alex traveled 147.2 miles.

How do I solve this question.

Answers

Hello, the solution of the geometry problem is in this photo.

AOC = 136°

BOC = 44°

3a+4+a=180
a=44

AOC=3*44+4
=136

COB=44

Help me asap! I will give you marks

Answers

Recall the binomial theorem.

[tex](a+b)^n = \displaystyle \sum_{k=0}^n \binom nk a^{n-k} b^k[/tex]

1. The binomial expansion of [tex]\left(1+\frac x3\right)^7[/tex] is

[tex]\left(1 + \dfrac x3\right)^7 = \displaystyle\sum_{k=0}^7 \binom 7k 1^{7-k} \left(\frac x3\right)^k = \sum_{k=0}^7 \binom 7k \frac{x^k}{3^k}[/tex]

Observe that

[tex]k = 1 \implies \dbinom 71 \left(\dfrac x3\right)^1 = \dfrac73 x[/tex]

[tex]k = 2 \implies \dbinom 72 \left(\dfrac x3\right)^2 = \dfrac73 x^2[/tex]

When we multiply these by [tex]8-9x[/tex],

• [tex]8[/tex] and [tex]\frac73 x^2[/tex] combine to make [tex]\frac{56}3 x^2[/tex]

• [tex]-9x[/tex] and [tex]\frac73 x[/tex] combine to make [tex]-\frac{63}3 x^2 = -21x^2[/tex]

and the sum of these terms is

[tex]\dfrac{56}3 x^2 - 21x^2 = \boxed{-\dfrac73 x^2}[/tex]

2. The binomial expansion is

[tex]\left(2a - \dfrac b2\right)^8 = \displaystyle \sum_{k=0}^8 \binom 8k (2a)^{8-k} \left(-\frac b2\right)^k = \sum_{k=0}^8 \binom 8k 2^{8-2k} a^{8-k} b^k[/tex]

We get the [tex]a^6b^2[/tex] term when [tex]k=2[/tex] :

[tex]k=2 \implies \dbinom 82 2^{8-2\cdot2} a^{8-2} b^2 = 28 \cdot2^4 a^6 b^2 = \boxed{448} \, a^6b^2[/tex]


The image of the point P(-3, 2) under the translation (2/1) is

Answers

The image of the point P(-3, 2) under the translation T<2,1> is (-1, 3)

How to determine the image of the translation?

The coordinate point is given as:

P = (-3, 2)

The translation is given as

T<2,1>

The translation can be represented as:

(x, y) = (x + 2, y + 1)

So, we have the following equation

P' = (-3 + 2, 2 + 1)

Evaluate the sum

P' = (-1, 3)

Hence, the image of the point P(-3, 2) under the translation T<2,1> is (-1, 3)

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if 7 cosec^2 theta-9 cot^2theta=7 then what is the value of tantheta​

Answers

[tex]\displaystyle\\Answer:\theta=\frac{\pi }{2}+\pi n.[/tex]

Step-by-step explanation:

[tex]\displaystyle\\7*cosec^2\theta-9*cot^2\theta=7\\7-7*cosec^2\theta+9*cot^2\theta=0\\7-\frac{7}{sin^2\theta}+9*\frac{cos^2\theta}{sin^2\theta} } =0\\\frac{7*sin^2\theta-7+9*cos^2\theta}{sin^2\theta} =0\\\frac{-7*(1-sin^2\theta)+9*cos^2\theta}{sin^2\theta} =0\\\frac{-7*cos^2\theta+9*cos^2\theta}{sin^2\theta} =0\\\frac{2*cos^2\theta}{sin^2\theta}=0\\[/tex]

[tex]2*cot^2\theta=0\\Divide\ the\ right\ and\ initial\ parts\ by\ 2:\\cot^2\theta=0\\cot\theta=0\\\theta=\frac{\pi }{2}+\pi n\ \ \ (n=0,\ 1,\ 2,\ 3\ ...).[/tex]

Which equation represents y = −x2 + 4x − 1 in vertex form?

Answers

The vertex form of the equation y = -x^2 + 4x - 1 is y = -(x - 2)^2 + 3

How to determine the vertex form of the quadratic equation?

The quadratic equation is given as:

y = -x^2 + 4x - 1

Differentiate the above quadratic equation.

This is done with respect to x by first derivative

So, we have:

y' = -2x + 4

Set the derivative to 0

-2x + 4 = 0

Subtract 4 from both sides of the equation

-2x + 4 - 4 = 0 - 4

Evaluate the difference in the above equation

-2x = -4

Divide both sides of the above equation by -2

x = 2

Rewrite as

h = 2

Substitute 2 for x in the equation y = -x^2 + 4x - 1

y = -2^2 + 4 *2 - 1

Evaluate the equation

y = 3

Rewrite as:

k = 3

A quadratic equation in vertex form is represented as:

y = a(x - h)^2 + k

So, we have:

y = a(x - 2)^2 + 3

In the equation y = -x^2 + 4x - 1, a = -1

So, we have:

y = -(x - 2)^2 + 3

Hence, the vertex form of the equation y = -x^2 + 4x - 1 is y = -(x - 2)^2 + 3

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1. How do you prove congruence through transformations?

2. How do you prove triangle congruence using congruency postulates? Give a general explanation of what the S and A stand for. Please name each of the postulates and what the letters stand for.

Answers

One can prove congruence through transformation if they have the same shape and size.

The congruency postulates include:

SSS - Side-Side-SideSAS - Side-Angle-SideASA- Angle-Side-AngleAAS - Angle-Angle-SideRHS - Right angle-Hypotenuse-Side

What is congruence?

In geometry, it should be noted that two figures are congruent if they have the same shape and size.

In this case, if two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

One can prove triangle congruence using congruency postulates by using the SSS theorem( side side side theorem).

It should be noted that the congruence postulate is used to illustrate that the triangles are equal.

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