Answer:
I replaced alpha by A and bitta by B. Sorry for this mistake
Answer:
4x² - 13x + 8 = 04x² - 11x + 5 = 016x² - 41x + 1 = 0x² + 5x + 4 = 0x² - 66x + 64 = 0Step-by-step explanation:
Given
α and β are roots of 4x²-5x-1=0Then the sum and product of the roots are:
α+b = -(-5)/4 = 5/4αβ = -1/4(i) Roots are α + 1 and β + 1, then we have:
(x - (α + 1))(x - (β + 1)) = 0(x - α - 1)(x - β - 1) = 0x² - (α+β+2)x + α+β+ αβ + 1 = 0x² - (5/4+2)x +5/4 - 1/4 + 1 = 0x² - 13/4x + 2= 04x² - 13x + 8 = 0(ii) Roots are 2 - α and 2 - β, then we have:
(x + α - 2)(x + β - 2) = 0x² + (a + β - 4)x - 2(α + β) + αβ + 4 = 0x² + (5/4 - 4)x - 2(5/4) - 1/4 + 4 = 0x² - 11/4x - 10/4 - 1/4 + 16/4 = 0x² - 11/4x + 5/4x = 04x² - 11x + 5 = 0(iii) Roots are α² and β², then:
(x - α²)(x-β²) = 0x² -(α²+β²)x + (αβ)² = 0x² - ((α+β)² - 2αβ)x + (-1/4)² = 0x² - ((5/4)² -2(-1/4))x + 1/16 = 0x² - ( 25/16 + 1/2)x + 1/16 = 0x² - 33/16x + 1/16 = 016x² - 33x + 1 = 0(iv) Roots are 1/α and 1/β, then:
(x - 1/α)(x - 1/β) = 0x² - (1/α+1/β)x + 1/αβ = 0x² - ((α+β)/αβ)x + 1/αβ = 0x² - (5/4)/(-1/4)x - 1/(-1/4) = 0x² + 5x + 4 = 0(v) Roots are 2/α² and 2/β², then:
(x - 2/α²)(x - 2/β²) = 0x² - (2/α² + 2/β²)x + 4/(αβ)² = 0x² - 2((α+β)² - 2αβ)/(αβ)²)x + 4/(αβ)² = 0x² - 2((5/4)² - 2(-1/4))/(-1/4)²x + 4/(-1/4)² = 0x² - 2(25/16 + 8/16)/(1/16)x + 4(16) = 0x² - 2(33)x + 64 = 0x² - 66x + 64 = 0A carpenter use 65 shelves to make 13 bookcases. She uses the same number of shelves for each bookcase. Are 32 shelves enough to build 6 more bookcases?
Answer:
Yes.
Step-by-step explanation:
First, find the unit rate for 1 bookcase.
65 ÷ by 13 = 5 (unit rate)
Then, multiply the unit rate by the bookcases the carpenter wants to make.
6 · 5 = 30
Finally, subtract the shelves the carpenter wants to use by 30.
32 - 30 = 2 shelves.
Therefore, 32 shelves is enough to build 6 more bookcases. <3
Gloria pays her insurance three times each year. The first payment is 50% of the annual
premium, and each of the next two payments is 25% of the annual premium. If the annual premium is
$1024, how much is each payment?
Answer:
First payment (50%) would be 512$
The next two payments would be 32$ each
Explanation:
.5(1024)= 512
.25(512)=128
.25(128)= 32
2x+3y = 17
2y + z = 1
5x +10z = 25
Find x
Answer:
x = 7
Step-by-step explanation:
Given the equations
2x + 3y = 17 → (1)
2y + z = 1 → (2)
5x + 10z = 25 → (3)
Rearrange (2) expressing z in terms of y
z = 1 - 2y
Substitute z = 1 - 2y into (3)
5x + 10(1 - 2y) = 25
5x + 10 - 20y = 25 ( subtract 10 from both sides )
5x - 20y = 15 → (4)
Multiply (1) by 20 and (4) by 3, then add to eliminate the y- term
40x + 60y = 340 → (5)
15x - 60y = 45 → (6)
Add (5) and (6) term by term to eliminate y, that is
55x = 385 ( divide both sides by 55 )
x = 7
Answer:
[tex]x = 7[/tex]
Step-by-step explanation:
From the equation [tex]2y + z = 1[/tex] , we can separate [tex]z[/tex] so that [tex]z = 1-2y[/tex].
The equation [tex]5x+10z = 25[/tex] can then be turned into:
[tex]5x+10(1-2y) = 25[/tex]
[tex]5x+10-20y = 25[/tex]
[tex]5x -20y = 15[/tex]
[tex]x-4y = 3[/tex] (divide by [tex]5[/tex] on both sides)
We can now use elimination to solve for one of the variable:
[tex]2(x-4y) = 2\times3[/tex] (times 2 to get [tex]2x[/tex] for elimination)
[tex]2x-8y=6[/tex]
[tex]2x+3y-(2x-8y) = 17-6[/tex]
[tex]11y = 11[/tex]
[tex]y = 1[/tex]
Now, we can use substitution to solve for [tex]x[/tex]:
[tex]2x+3y = 17[/tex]
[tex]2x+3 = 17[/tex]
[tex]x = 7[/tex]
Hope this helps :)
What is 400,000 in scientific notation?
O A) 40 x 104
OB) 400 x 103
OC) 4 x 106
OD) 4 x 105
Answer:
4 x 10⁵
so that i think will be OD)
what are the three products of celullar respiration
Answer:
Cellular respiration is this process in which oxygen and glucose are used to create ATP, carbon dioxide, and water. ATP, carbon dioxide, and water are all products of this process because they are what is created.
Step-by-step explanation:
what does x = ? need help with the pythagorean theorem
Answer:
The answer is 9
Step-by-step explanation:
set up the equation as 12^2 + b^2 = 15^2
solve the exponents and you get 144 + b^2 = 225
Then subtract 144 from both sides and get 81
After that just do the square root of 81 and you get 9
pls mark brainliest
PLEASE HELP!!!!! 2.) What is the perimeter of a polygon with vertices at (-3, 1), (5, 1), (-3, 4), (5, 4)? 30 points if you can help
Answer:
22 units
Step-by-step explanation:
Known Quantities:
AB = CD and AC = BDCalculations:
AB = 8AC = 3CD = 8BD = 3Final Calculations:
perimeter = 2 x (3+8)perimeter = 227. What is the mass of 20.6cm3 of iron if iron
has a density of 7.87g/cm3?
A. 162.1 g/cm3
B. 162.1 cm3
C. 162.1 g/mL
D. 162.1 g
Answer:
Option B
Step-by-step explanation:
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Can someone help pls
Find the area of the shape below
Answer:
????
Step-by-step explanation:
Answer:
where's the shape
Step-by-step explanation:
what's the length and width, maybe I can help
Every week, Hector’s 20 hours earns $210.00. He earns a constant amount of money per hour.
Part B
Write an equation that can be used to determine Hector's earnings, in dollars, m, for h hours of work.
Answer: 20 weeks in 1 week
in 2 weeks 40
Step-by-step explanation:
This proves that the equation is true
Can someone help me out with this please!
What is the domain and range of the function: y= 10(4) + 100
Domain: ____
Range: ____
1. The sum of a number and ten.
Answer:
x+10
Step-by-step explanation:
sum = addition (+)
number = variable because we don't know what the number is yet (x)
ten=ten
So by this we can get x+10
The length of a rectangle is represented by 4 + 6x. The width is half the length. What expression represents the perimeter of the rectangle? Explain your reasoning
Answer:
12+18x is the perimeter
Step-by-step explanation:
4+6x+4+6x+1/2(4+6x)+1/2(4+6x)=
3(4+6x)=
12+18x
A number x is greater than 3.
Answer:
4,5,6,7,8,9..........................
Heather has a bag with 3 marbles, 7 tennis balls, 3 apples, and 3 books. What is the ratio of books to tennis balls?
A.
(3/7) or 3:7
B.
(7/8) or 7:8
C.
(4/7) or 4:7
Answer:
B
Step-by-step explanation:
I hope it's correct and I'm thinking it is,. Sorry if it's not
Answer:
I think A
Step-by-step explanation:
3/7. Or 3:7
Cause there are 3 books and 7 tennis balls
Find two integers whose sum is 19 and product is 90
Answer:
Step-by-step explanation:
The two integers are 10 and 9
10 + 9 = 19
10 x 9 = 90
The two numbers are 9 and 10.
What are integers?A number without a decimal or fractional element is known as an integer, which can be both positive and negative, including zero
Let the two integers be x and y.
Given that, the sum of the integers is 19, therefore,
x + y = 19
x = 19 - y
Given that, the product of the integer is 90:
xy = 90
(19-y)y = 90
19y - y² = 90
y² -19y + 90 =0
y² -10y - 9y + 90 = 0
y(y - 10) -9(y - 10) = 0
(y -9)(y - 10) = 0
y = 9 and y = 10
Hence, the two numbers are 9 and 10.
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A small town has two local high schools. High School A currently has 850 students and is projected to grow by 35 students each year. High School B currently has 700 students and is projected to grow by 60 students each year. Let AA represent the number of students in High School A in tt years, and let BB represent the number of students in High School B after tt years. Write an equation for each situation, in terms of t,t, and determine after how many years, t,t, the number of students in both high schools would be the same.
An equation for each situation, in terms of t, is y = 850e^35t and
y = 700e^70t
The required equation will be in the form y = Ae^kt
where:
k is the growth constantA represents the number of students in High School A in t years.B represents the number of students in High School B after t years.If High School A currently has 850 students and is projected to grow by 35 students each year, hence;
A = 850k = 35 (growth factor)Substituting into the formula, we will have:
y = 850e^35t
If High School B currently has 700 students and is projected to grow by 60 students each year, hence;
A = 700k = 60 (growth factor)Substituting into the formula, we will have:
y = 700e^70t
An equation for each situation, in terms of t, is y = 850e^35t and
y = 700e^70t
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A mosquito is walking at random on the nonnegative number line. She starts at $1$. When she is at $0$, she always takes a step $1$ unit to the right, but, from any positive position on the line, she randomly moves left or right $1$ unit with equal probability. What is the expected number of times the mosquito will visit $0$ before the first time she visits $4$?
The expected number of times the mosquito will visit 0 before the first time she visits 4 is 3/4 by using concept of probability and geometric series.
Given that repeatedly throwing a fair tetrahedral die, trying to get a 4 (success) before getting a number from 1 to 3 (failure), which follows a geometric distribution.
To find the expected number of visits to 0 before reaching 4, analyze two cases:
success (reaching 4 before 0) and failure (reaching 0 before 4).By the given data, the probabilities of two cases are:
P(success) p = 1/4,
P(failure)=3/4.
The expected count of failures, X, can be found in terms of success probability p as
[tex]E[X]=\sum_{x=0}^{+\infty}[x(1-p)^xp][/tex]
On substituting p = 1/4 gives:
[tex]E(X) = 0(1/4)+1(3/4)(1/4)+2(3/4)^2(1/4)+3(3/4)^3(1/4)+.........[/tex]
On taking (1/4) gives
[tex]E(X) = 1/4[3/4+(3/4)^2+(3/4)^3+......+\infty]----(1)[/tex]
Consider [tex]3/4+(3/4)^2+(3/4)^3+......+\infty[/tex]
This forms a geometric series tends to infinity and the sum is given by :
[tex]S_\infty = a/(1-r)[/tex]
Here: a= 3/4
r = 3/4
Plugging these values into formula gives:
[tex]3/4+(3/4)^2+(3/4)^3+......+\infty = S_\infty[/tex]
[tex]S_\infty= 3/4/(1-3/4)[/tex]
On subtracting the denominator gives:
= 3/4/(1/4)
On taking reciprocal gives:
= 3/4 * 4
On simplifying gives:
= 3
Substituting these value into given equation 1 gives:
E(X) = 1/4*[3]
On multiplying gives:
E(X) = 3/4.
Therefore, the expected number of times the mosquito will visit 0 before the first time she visits 4 is 3/4 by using concept of probability and geometric series.
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A cuboid with a volume of 925cm^3 has dimensions.
4cm, (x + 1) cm and (x + 11) cm.
Show clearly that x^2 + 12x - 220 = 0
Solve the equation by factorising, making sure you show the factorisation.
State both values of x on the same line.
Finally, find the dimensions of the cuboid, writing all three on one line.
Answer:
Step-by-step explanation:
Volume of the cuboid = Length * Width * Height
Given
Length = 4cm
Width = (x+1)cm
Height = (x-11)cm
Volume of the cuboid = 4(x+1)(x+11)
Volume of the cuboid = 4(x^2+11x+x+11)
Volume of the cuboid = 4(x^2+12x+11)
Volume of the cuboid = 4x*2+48x+44
925 = 4x*2+48x+44
4x*2+48x+44-925 = 0
4x*2+48x-881 = 0
Divide through by 4
x^2 + 12x - 220.25 = 0
Factorize using the general formula;
x = -12±√12²-4(-220.25)/2
x = -12±√144+881/2
x = -12±√1025/2
x = -12±32/2
x = 12+32/2
x = 20/2
x = 10
Hence the dimension of the cuboid is 4cm, (10+1)cm and (10+11)cm
Dimension is 4cm by 11cm by 21cm
Help Really quick
The table shows data collected from 14 people.
Shoe Size, x 8.5 9 9 9.5 10 10 10.5 10.5 11 11 11 12 12 12.5
Height (inches), y 66 68.5 67.5 70 70 72 71.5 70 71 71.5 73 73.5 74 74
Determine the correlation coefficient for the data. Round to the nearest hundredth.
r=
Answer:
i need the answer ;(
Step-by-step explanation:
Is the relation in the mapping below a function? Explain.
Domain
1,3,5,7
Range
2,4,6
Answer: No
Step-by-step explanation:
The range and the Domain aren’t equal
Find the slope of the line that contains (1, 6) and (10.-9).
Answer:
-5/3
Step-by-step explanation:
m=−9−6/10−1
m=−15/9
Slope=
−15/9=−5/3
HELP ASAP OTS DUE IN 5 MINUTES!!!!
Find the slope of the line.
0-5
0 -1
O 1
None of the above
Answer:
0 -1
Step-by-step explanation:
because the line is going down which is negative
i need the answer for this question
Answer:
x=5
y=7
(5,7)
Step-by-step explanation:
solved by substitution.
Starting from the same place a plane took 1 hr longer to fly 705 mi to Atlanta, than to fly 470 mi to New Orleans. If the plane flew at the same rate for the entire trip, what was the flying time for the shorter leg? a. 1 2 hr c. 1 1 2 hrs b. 1hr d. 2 hrs
Answer: d. 2 hrs
Step-by-step explanation:
Let x = Time taken(in hours) to fly 470 mi to New Orleans. (Shorter leg)
Then Time taken to fly 705 mi to Atlanta= x+1 (longer leg)
Since the plane flew at the same rate for the entire trip.
i.e. Speed 705 mi to Atlanta = Speed to fly to New orleans
[tex]\Rightarrow\ \dfrac{705}{x+1}=\dfrac{470}{x}\ \ \ \ \ \ \ [\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}]\\\\\Rightarrow\ 705x=470(x+1)\\\\\Rightarrow\ 705x=470x+470\\\\\Rightarrow\ 705x-470x=470 \\\\\Rightarrow\ 235x=470\\\\\Rightarrow\ x=\dfrac{470}{235} \\\\\Rightarrow\ x=2[/tex]
hence, the flying time for the shorter leg = 2 hours
So (d) is the correct option.
What is the answer to this
2(x-5)-6=-18
x=___
Answer:
x= -1
Step-by-step explanation:
Distribute:2(x -5) -6 = -18
2x -10 -6 =-18
Subtract the numbers:2x -10 -6 =-18
2x -16 = -18
Add 16 to both sides of the equation:2x =16 += -18
2x -16 +16 = -18 +16
Simplify:2x = -2
Divide both sides of the equation by the same term:2x/2 = -2/2
Simplify:x=-1
Answer:x = -1
-3y-2=7 what is the value of y
Answer:
y = -3
Step-by-step explanation:
Add 2 to each side to isolate the -3y. Doing this leaves you with -3y=9. You then divide each side leaving you with y = -3
Answer:
The equation has one solution at which y = -3.
Step-by-step explanation:
To find the value of y, we need to isolate it from the rest of the equation.
Add 2 to both sides of the equation.Divide by -3 on both sides of the equation.Simplify the equation if necessary.[tex]\displaystyle{-3y-2=7}\\\\-3y = 9\\\\\bold{y = -3}[/tex]
We can check to make sure that y = -3 by substitution.
[tex]\displaystyle{-3(-3)-2 = 7}\\\\9-2=7\\\\7= 7 \ \checkmark[/tex]
Because we got a true statement, we can determine that y = -3.
A rock is thrown from a cliff into a ravine. The function ℎ() = −16^2 + 19 + 2560 describes the height, in feet, of the rock from the ground seconds after it is thrown. What is the height of the rock, in feet, 2 seconds after it is thrown?
Answer:
d
Step-by-step explanation:
I need help someone please
Answer: [tex]\frac{6}{2}[/tex]
Step-by-step explanation:
1. Use this [tex]\frac{f(b)-f(a)}{b-a}[/tex] to find the average rate of change
2. Average rate of change = [tex]\frac{f(8)-f(6)}{8-6}[/tex]
3. [tex]f(8) = - 4[/tex] and [tex]f(6) = -10[/tex] (To find these you will need to look at the points on the graph. So the points (8,-4) and (6,-10))
4. Replace -4 to f(8) and -10 to f(6)
5. So it will look like[tex]\frac{-4-(-10)}{8-6}[/tex]
6. Solve [tex]\frac{6}{2}[/tex]