From Vieta's formulas, we have
[tex]3x^2 - 4x + 1 = 3 (x - \alpha) (x - \beta) \\\\ ~~~~~~~~~~~~~~~~~ = 3x^2 - 3 (\alpha + \beta) + 3\alpha\beta \\\\ \implies \begin{cases} \boxed{\alpha + \beta = \frac43} \\\\ \boxed{\alpha\beta = \frac13} \end{cases}[/tex]
a) Note that
[tex](\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2 \\\\ \implies \alpha^2 - \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta - 2\beta^2[/tex]
We have exact values for [tex]\alpha+\beta[/tex] and [tex]2\alpha\beta[/tex], but sadly not for [tex]-2\beta^2[/tex]. This means we'll need to solve for [tex]\beta[/tex] explicitly.
By factorizing, we have
[tex]3x^2 - 4x + 1 = (3x - 1) (x - 1) = 0 \\\\ \implies x = \dfrac13 \text{ or } x = 1[/tex]
Then the value of [tex]\alpha^2-\beta^2[/tex] depends on which is chosen to be the larger root, and
[tex]\alpha^2 - \beta^2 = \dfrac1{3^2} - 1^2 = \boxed{-\dfrac89} \text{ or } \alpha^2 - \beta^2 = 1^2 - \dfrac1{3^2} = \boxed{\dfrac89}[/tex]
While we're at it, we can also immediately find
[tex]\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\dfrac43\right)^2 - 2\cdot\dfrac13 = \dfrac{10}9[/tex]
b) Combine fractions to get
[tex]\dfrac2{\alpha^2} + \dfrac2{\beta^2} = \dfrac{2\beta^2 + 2\alpha^2}{\alpha^2\beta^2} \\\\ ~~~~~~~~~~~~~ = 2 \cdot \dfrac{\alpha^2 + \beta^2}{(\alpha\beta)^2} \\\\ ~~~~~~~~~~~~~ = 2\cdot \dfrac{\frac{10}9}{\left(\frac13\right)^2} \\\\ ~~~~~~~~~~~~~ = \boxed{20}[/tex]
c) Combine fractions again.
[tex]\dfrac2{\alpha^2} - \dfrac2{\beta^2} = 2 \cdot \dfrac{\beta^2 - \alpha^2}{(\alpha\beta)^2} \\\\ ~~~~~~~~~~~~~ = -2\cdot \dfrac{\pm\frac89}{\left(\frac13\right)^2} \\\\ ~~~~~~~~~~~~~ = \boxed{\pm16}[/tex]
The second side of a triangular deck is 3 feet longer than the shortest side, and the third side is 3 feet shorter than twice the length of the shortest side. If the perimeter of the deck is 52 feet, what are the lengths of the three sides?
The shortest side of the triangle measures 17 feet.The second side of the triangle measures 20 feet.The third side of the triangle measures 31 feet.What is the length of the three sides?The perimeter of a triangle is the sum of the length of the three sides of the triangle. Let the shortest side be represented with s.The second side = 3 + s The third side = 2s - 3 2s - 3 + 3 + s + s = 68 4s = 68s = 17The second side = 17 + 3 = 20 The third side = = (2 x 17) - 3 = 31
66 between numbers 4 and 39 put in four numbers, such that arithmetic sequence would be done. what is the third term of this arithmetic sequence?
The third term of the arithmetic sequence is 18.
How to solve arithmetic sequence?The arithmetic sequence formula can be represented as follows:
nth term = a + (n - 1)d
where
a = first termd = common differencen = number of termsTherefore, 4 numbers are inserted between 4 and 39.
Hence,
a = 4
39 = 4 + 5d
39 - 4 = 5d
35 = 5d
d = 35 / 5
d = 7
Hence, the common difference of the arithmetic sequence is 7.
Therefore, the sequence of the arithmetic sequence is as follows:
4, 11, 18, 25, 32, 39
Therefore, the third term of the arithmetic sequence is 18
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A walking tour of Washington, D.C., costs $20 per person. Show that the relationship between the number of people in a tour group and the total cost is a proportional relationship. Write an equation for the relationship.
The Total cost for a tour group consisting of n number of people is 20n
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let T represent the total cost if a tour group consisting of n people
Since A walking tour of Washington, D.C., costs $20 per person. Hence, for n number of people:
Total cost = $20 per person * n persons = 20n
The Total cost for a tour group consisting of n number of people is 20n
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its a data handling question can u guys please help
(i) Maximum students were watching T.V for 4-5 hours.
(ii) 34 students watched T.V for less than 4 hours.
(iii) 14 students spent more than 5 hours watching TV.
Given from the graph we can say that:
The process of acquiring, storing, and displaying data in a way that makes it easy for others to use (read or fetch data), such as in graphs or charts, is known as data handling. Statistics is another name for it. Additionally, it is employed for data comparison and mean, median, and mode extraction.
(i) maximum number of students watching TV =32
Almost 32 students watched TV for 4-5 hours.
(ii) For students watching TV for less than 4 hours = ?
= 4 ₊ 8 ₊ 22
= 34
(iii) students who spent more than 5 hours watching TV = ?
= 8 ₊ 6
= 14
Hence we get the desired answers.
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n a triangle, the ratio of the measures of the sides is 2: 2: 3 and the perimeter is 392 inches. Find the length of the longest side of the triangle.
Answer:
168 inches
Step-by-step explanation:
Patrice is trying a low-calorie diet. She would like to keep her calories consumed between the levels shown in the following compound inequality: 2450 < 2x + 50 and 2x + 50 < 3250. Solve for x in the inequality, and explain what the answer represents.
The solution to the compound inequality is 1200 < x < 1600 and the meaning is she must keep her calories consumed between 1200 and 1600
How to solve for x in the inequality, and explain what the answer represents?The compound inequality is given as
2450 < 2x + 50 and 2x + 50 < 3250
Subtract 50 from both sides of the inequality
So, we have
2400 < 2x and 2x < 3200
Divide through the inequality by 20
So, we have
1200 < x and x < 1600
Combine the inequalities
1200 < x < 1600
Hence, the solution to the compound inequality is 1200 < x < 1600 and the meaning is she must keep her calories consumed between 1200 and 1600
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I need help with introduction to probability lesson. Photo is attached below.
The probability of getting heads is 31/50.
How to calculate the probability?Number of tosses = 50
Number of heads = 31
The probability of getting heads will be:
= Number of heads / Total number of tosses
= 31/50
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A group of friends wants to go to the amusement park. They have no more than $325
to spend on parking and admission. Parking is $13, and tickets cost $39 per person,
including tax. Write and solve an inequality which can be used to determine p, the
number of people who can go to the amusement park.
Mr. Wise is ordering a set of books for his class. Each book costs $6. There is a flat shipping fee of $5. The table below shows the relationship between the total cost, c, and the number of books, b. Fill in the blanks to complete the table. Number of Book, b 2 [a] 6 [b] 10 Total Cost ($), c 17 29 [c] 53 65
Filling the table to show the relationship between the total cost, c, and the number of books ordered is as follows:
Number of Books Total Cost ($)
2 $17 ($6 x 2 + $5)
6 $41 ($6 x 6 + $5)
10 $65 ($6 x 2 + $5)
What is the total cost?The total cost is a function of the number of units and the variable cost per unit plus the fixed costs.
For instance, the total cost of 9 books, in this scenario, is $59 ($6 x 9 + $5).
Data and Calculations:Cost per book = $6
Shipping fee = $5
Number of Books Total Cost ($)
2 $17 ($6 x 2 + $5)
6 $41 ($6 x 6 + $5)
10 $65 ($6 x 2 + $5)
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Sorry if that's hard to see, but I need ur help!
The image represents what geometric construction?
A Parallel Lines Construction
B Copy an Angle Construction
C Angle Bisector Construction
D Perpendicular Bisector Construction
Please help me :3 please explain
Using a proportional relationship, the amounts are given as follows:
Tennis 3, Soccer 7.Tennis 36, Soccer 84.What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality.
For this problem, we have that:
The input variable is the number of tennis players.The output variable is the number of soccer players.From the first row of the table, the constant is given as follows:
k = 35/15 = 7/3.
Hence the relationship is:
y = 7/3x.
For the second row of the table, we have that x = 3, hence:
y = 7/3 x 3 = 7.
For the third row of the table, we have that y = 84, hence:
84 = 7/3x
x = 84 x 3/7
x = 36.
Then the amounts are given as follows:
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Find the mean of the data set: 9, 6, 7, 3, 10, 9
A. 6
B. 7
C. 8
D. 9
Answer: 7
Step-by-step explanation: add all the numbers up and divide them by how many numbers they are
what is -14/64 as a decimal ?
Hi ~
︵‿︵‿︵‿︵︵‿︵‿︵‿︵︵‿︵‿︵‿︵︵‿︵‿︵‿︵︵‿︵‿︵
-14/64
⚜ E x p l a n t i o n : -Divide the numbers: [tex]\bold{14}[/tex] ÷ [tex]\bold{64 = 0.21875}[/tex]
[tex]\bold{= -0.21875}[/tex]
Hope It Helps...
[tex]\textsc{A \: n \: s \: w \: e \: r:}[/tex]
TheAestheticGirl
Find the volume of a cube that has a total surface area of 54 square millimeters.
The volume of a cube that has a total surface area of 54 square millimeters is 27 cubic millimeters
How to determine the volume?The surface area is given as
A = 54
This is calculated using
A = 6L^2
So, we have
6L^2 = 54
Divide
L^2 = 9
This gives
L = 3
The volume is
V = L^3
This gives
V = 3^3
Evaluate
V = 27
Hence, the volume of a cube that has a total surface area of 54 square millimeters is 27 cubic millimeters
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The width of a rectangle is the length minus 5 units. The area of the rectangle is 24
units. What is the length, in units, of the rectangle?
Answer:
L = 8
w = 3
Step-by-step explanation:
Givens
w = L - 5
L = x
Area = 24
Formula
Area = L * w
Solution
Area = (x) * (x - 5)
24 = x * (x - 5) Remove the brackets
24 = x^2 - 5x Subtract 24 from both sides
0 = x^2 - 5x - 24 This will factor
0 = (x - 8)(x + 3)
x + 3 is an extraneous root. You cannot have a length of -3 units no matter what the units are.
x - 8 = 0
x = 8
Answer
The length = 8
The width is = 8 - 5 = 3
Solve 4∣x+5∣=24
A.x = –11 and x = –1
B.x = 11 and x = –1
C.x = –11 and x = 1
D.x = 11 and x = –11
Step-by-step explanation:
You first need to get the absolute vaule of x+5 by its self. So you need to divide 4 on yet sides
[tex] |x + 5| = 6[/tex]
Then set up two equations a negative and a postive on
[tex] |x + 5| = 6[/tex]
x=1
[tex] |x + 5| = - 6[/tex]
x=-11
Hope that helped!
Feel free to ask questions!
Alessa had 50 in her account. she then received four $-25 charges to her account. what is the new balance of her account?
i need help!! ……………….
Answer:
Step-by-step explanation:
The first one its easy
simplify (4a-3)(a-9)=4a² -39a+27
Answer:
Step-by-step explanation:
0=0
Help please!
I also need an explaination on how to do it.
Which expression is equivalent to −4 + 9 − 2x − 9x?
−11x + 5
−11x + 13
−11x − 5
11x + 13
Answer:
−11x + 5Step-by-step explanation:
−4 + 9 − 2x −9x
−4 + 9 + − 2x + −9x
Combine Like Terms:−4 + 9 + −2x +− 9x
( −2x+−9x )+( −4+9 )
−11x + 5
Describe how you would graph the inequality x ≥−13
Pls help me
The principal is measuring the floor of the second-grade classroom.
What is the best estimate for the length of a classroom?
The best measure to do so will be is in feet that is the principal will measure it as 10 feet.
The principal has to measure the floor of the second grade classroom.
The best measure to do so will be is in feet
As centimeters and inches will be too short to measure it and can cause problems while further maintaining the records.
Yards on the other hand is a too big unit of measurement to bthe length of the classroom.
Therefore, the best measure to do so will be is in feet that is the principal will measure it as 10 feet.
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Your question was incomplete. Please refer the content below
The principal is measuring the floor of the second-grade classroom.
What is the best estimate for the length of a classroom?
10 yards
10 centimeters.
10 feet
10 inches
-1/4 x-6 =2 what is x equal to
Answer:
x=−32Step-by-step explanation:
−1/4x−6=2
Add 6 to both sides:−1/4x−6+6=2+6
−1/4x=8
Multiply both sides by 4/(-1):(4−1)*(−14x)=(4−1)*(8)
x=−32
Answer: x=−32
−1/4x−6=2
Add 6 to both sides:
−1/4x−6+6=2+6
−1/4x=8
Multiply both sides by 4/(-1):
(4−1)*(−14x)=(4−1)*(8)
x=−32
Step-by-step explanation: hope this helps!!!
2 divided by $1.03 round to the nearest cent ?
Answer:
.05
Step-by-step explanation:
Select the correct answer.
Consider figures 1 and 2 shown in the coordinate plane. Figure 1 has been transformed to produce figure 2.
-00
55-
N
2
N
ON
6
80
Which notation describes this transformation?
OA. (x,y) = (x,-)
OB. (x,y)=(x-4,y - 8)
OC. (x,y) = (x - 8,y - 8)
OD. (x,y)= (-2,-y)
9.
8.
Using translation concepts, the notation that describes this transformation is:
C. (x,y) -> (x - 8, y - 8).
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
To generate figure 2 from figure 1, these following translations happened:
8 units left, hence x -> x - 8.8 units down, hence y -> y - 8.Thus the rule is given by:
C. (x,y) -> (x - 8, y - 8).
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Answer: (x',y') = (x - 4,y - 8)
Step-by-step explanation:
28. Use the properties of the mean and median to determine which
are the correct mean and median for the following histogram.
(i) Mean is 3.6; median is 4.8
(ii) Mean is 4.8; median is 3.6
(iii) Mean is 3.5; median is 3.1
(iv) Mean is 4.2; median is 4.1
The properties of the mean and median to determine which
are the correct mean and median for the following histogram. The answer is Mean is 3.5; median is 3.1.
Mean
[tex]\mu=& \varepsilon x \cdot P(x)[/tex]
= 0(0.04)+1(0.08)+2(0.125)+3(0.24) +4(0.15)+5(0.14)+6(0.12)+7(0.05)
= 0+0.08+0.25+0.72+0.6+0.7+0.72+0.35
= 3.42 [Approx: 3.5]
Mean =3.5 of median 3.1.
A data set's mean (average) is calculated by summing all of the numbers in the set, then dividing by the total number of values in the set. When a data collection is ranked from least to greatest, the median is the midpoint. The number that appears most frequently in a data collection is called the mode.
The median is computed by sorting the scores into numerical order, dividing the total by 2, rounding the result up if the number of scores is odd to determine the median position, or by averaging the median position and the position after it, if the number of scores is even.
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while standing in front of the school, jace recorded the number of students who walked, rode their bikes, took the bus, and were driven there.
This is an observational study because this is taken as observing all possible ways.
What is observational study?An observational review is utilized to respond to an exploration question dependent simply upon what the scientist notices. There is no impedance or control of the examination subjects, and no control and treatment gatherings. These examinations are much of the time subjective in nature and can be utilized for both exploratory and logical examination purposes. In observational examinations, the analyst isn't permitted to control anything. They just notice and gather patterns inside the set up and hypothesize from those outcomes. Observational examination takes into consideration more precise information assortment, eliminating predispositions and inspecting blunders. Notwithstanding, it tends to be tedious because of extended periods of time of dormancy.
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Based on the what we established about the classification of y and using the closure of integers, what does the equation tell you about the type of numbers y must be for the product to be rational? What conclusion can you now make about the result of multiplying a rational and an irrational number
The statements are:
x . y = m/n
substituting the values.
a/b . y = m/n
multiplication property of equality.
b/a . a.b . y = m/n . b/a
simplify.
y = mb/na
The product of a rational number with an irrational number is an irrational number. To see this assume that x is a rational number and y an irrational number. Then let us assume that the product xy is rational, which means that there are integers a,b such that xy=a/b. But then we obtain y=(1/x)(a/b) which is also rational since the set of rational numbers is closed under multiplication. But this is a contradiction since y was assumed to be an irrational number.
Hence we made the required conclusion.
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The coordinate -1 has a weight of 1, the coordinate 4 has a weight of 2, and the coordinate 9 has a weight of 2. Find the
weighted average.
The weight average of the coordinates is 5
How to determine the weight average?The given parameters are:
Coordinate -1 has a weight of 1Coordinate 4 has a weight of 2Coordinate 9 has a weight of 2The weight average is then calculated as:
Weight average = Sum of (Weight * Coordinate)/Sum of Weights
So, we have:
Weight average = (-1 * 1 + 4 * 2 + 9 * 2)/(1 + 2 + 2)
Evaluate the quotient
Weight average = 5
Hence, the weight average of the coordinates is 5
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