The values of a and b are 8 and - 9.
Given that:
f(x) = x + a/x + b ........ (1)
f(1) = 0, f(2) = - 3
Putting the value of x = 1 in equation (1). We get,
f(1) = 1 + a/1 + b
f(1) = 1 + a + b ....... (2)
Putting the value of x = 2 in equation (1). We get,
f(2) = 2 + a/2 + b
f(2) = (4 + a + 2b)/2 ........ (3)
Putting the value of f(1) = 0 in equation (2). We get,
a + b = - 1 ......... (4)
Putting the value of f(2) = -3 in equation (3). We get,
=> -3 = (4 + a + 2b)/2
=> -6 = (4 + a + 2b)
=> a + 2b = - 10 ......... (5)
To find the value of a and b we have to solve equations (4) and (5) by elimination method.
So, for that subtract equation (4) from (5).
We got the value of b = - 9.
Then put the value of b in equation (4).
We got the value of a = 8.
Checking the value of a and b satisfies the value of the function's output and input.
Hence, the values of a and b are 8 and - 9 respectively.
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f(x) = (1 + =)",
k, n EN,
Given that the coefficient of x³ is twice the coefficient of x² in the binomial expansion of f(x),
(a) prove that n = 6k + 2.
Given also that the coefficients of x4 and x5 are equal and non-zero,
(b) form another equation in n and k and hence show that k = 2 and n = 14.
Using these values of k and n,
(c) expand f(x) in ascending powers of x, up to and including the term in x5.
Give each coefficient as an exact fraction in its lowest terms
n>2
Given that the coefficient of x³ is twice the coefficient of x² in the binomial expansion of f(x)
(a) It is proved that n = 6k + 2
(b)k = 2 and n=14
(c) After expansion, the equation is
1 + 14x + 91/4x^2 + 1001/2x^3 + 1001/16x^4 + 1001/16x^5
What is the binomial theorem?
The algebraic expansion of a binomial's powers is described by the binomial theorem (also known as binomial expansion) in elementary algebra.
The theorem states that the polynomial (x + y)n can be expanded into a sum including terms of the form axbyc, where the exponents b and c are nonnegative integers and b + c = n, and the coefficient an of each term is a particular positive integer dependent on n and b.
Solution Explained:
f(x)= (1+x/k)^n
Given that the coefficient of x^3 is twice the coefficient of x^2 in the binomial expansion of f(x)
a)
Coeff of x^3 = 1/6[n(n-1)(n-2)/k3]
Coeff of x^2 = 1/2[n(n-1)/k2]
1/6[n(n-1)(n-2)/k3] = 2* 1/2[n(n-1)/k2]
Canceling on both sides gives:
(n-2)/(6k) = 1
n = 6k+2 Hence proved
Given also that the coefficients of x^4 and x^5 are equal and non-zero,
b)
Coeff x^4 = 1/24 [n(n-1)(n-2)(n-3)/k4
Coeff x^5 = 1/120 [n(n-1)(n-2)(n-3)(n-4)/k5
1/120 [n(n-1)(n-2)(n-3)(n-4)/k5 = 1/24 [n(n-1)(n-2)(n-3)/k4
Canceling gives:
(n-4)/(120k) = 1/24
n = 5k+4
From previous:
n = 6k+2 = 5k+4
k = 2
n=14
Using these values of k and n,
c)
Give each coefficient as an exact fraction in its lowest terms
1 + 14x + 91/4x^2 + 1001/2x^3 + 1001/16x^4 + 1001/16x^5
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URGENT PLEASEEE!
What is an equation of the line that passes through the points (2, -8) and (-6, 0)
Answer:
It should be y=2x-1
Step-by-step explanation:
Calculate the area of the square!
Answer:
The answer is 169
Step-by-step explanation:
Find an equation for the perpendicular bisector of the line segment whose endpoints are (4,−1) and (8,7).
Answer: [tex]y=-\frac{1}{2}x+6[/tex]
Step-by-step explanation:
The slope of the line segment is [tex]\frac{-1-7}{4-8}=2[/tex].
So, the slope of the perpendicular bisector is -1/2.The midpoint of the line segment is [tex]\left(\frac{4+8}{2}, \frac{-1+7}{2} \right)=(6, 3)[/tex].
Thus, the equation is:
[tex]y-3=-\frac{1}{2}(x-6)\\\\y-3=-\frac{1}{2}x+3\\\\y=-\frac{1}{2}x+6[/tex]
HELP ME PLS IM STRUGGLING
1. If 1(-1) for the polynomial f(z) = 2z+z²-5 is-2, can you use the Factor Theorem to find the other factor?
No, because (x+1) is not a factor of the polynomial f(z) = 22 +2²-5,
Yes, because (x + 1) is a factor of all degree 4 polynomials.
No, because-2 is not an irrational number.
Yes, because-2 is the opposite of the leading coefficient of f(z) = 2x¹ +2²-5.
Answer: yes, because -2 is the opposite of the leading coefficient
Step-by-step explanation:
NAME
DATE
Topic Representing Linear Relationships
PERIOD
4. A mountain road is 5 miles long and gains elevation at a constant rate.
After 2 miles, the elevation is 5,500 feet above sea level.
After 4 miles, the elevation is 6,200 feet above sea level. (Lesson 3-6)
a. Find the elevation of the road at the point where the road begins.
The elevation of the road is 4800 feet.
From the question, we have
y = mx + b
where m is the rate of change, b is the initial value of y and y, x are variables.
Let y represent the elevation of the road after x miles.
After 2 miles, the elevation is 5500 feet above sea level.
5500 = 2m + b (1)
After 4 miles, the elevation is 6200 feet above sea level.
6200 = 4m + b (2)
Solving equation 1 and 2, we get
m = 350, b = 4800
The elevation of the road 4800 feet.
Subtraction:
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −. For example, there are nine oranges arranged as a stack (as shown in the above figure), out of which four oranges are transferred to a basket, then there will be 9 – 4 oranges left in the stack, i.e. five oranges. Therefore, the difference between 9 and 4 is 5, i.e., 9 − 4 = 5. Subtraction is not only applied to natural numbers but also can be incorporated for different types of numbers. The letter "-" stands for subtraction. Minuend, subtrahend, and difference are the three numerical components that make up the subtraction operation. A minuend is the first number in a subtraction process and is the number from which we subtract another integer in a subtraction phrase.
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help meeeeeeeeeeee pleaseeeeeeeeeee rn rnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
help meeeeeeeeeeee pleaseeeeeeeeeee rn rnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
The maximum area that the material would cover is 640, 000 square yards
What is rate of change?A rate of change is a rate that describes how one quantity changes in relation to another quantity. If x is the independent variable and y is the dependent variable, then
rate of change = change in y / change in x
if 3200 yards was used for fencing, then 3200 yards is perimeter
and perimeter of a rectangle = 2( L + B)
so that,
2(L + B) = 3200
L + B = 1600
express L in terms of B
L = 1600 - B
Area(A) = L X B
A = L X B
substitute L = 1600 -B
A = (1600 - B) x B
multiply out
A = 1600B - [tex]B^{2}[/tex]
for maximum area,
[tex]\frac{dA}{dB}[/tex] = 0
But dA / dB = 1600 - 2B
1600 - 2B = 0
2B =1600
B = 1600/2
B = 800 yards
L = 1600 - B
L = 1600 - 800
L = 800 yards
Area = L X B
A = 800 x 800
A = 640,000 square yards
In conclusion, he maximum area is 640,000 square yards and the dimensions are 800 yards by 800 yards
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67,107 rounded to the nearest ten-thousand
Answer:
70,000
Step-by-step explanation:
Round up if the next digit is higher than or equal to 5.
edit: accidentally answered twice
Answer:
70,000
Step-by-step explanation:
Round up if the next digit is higher than or equal to 5.
so what is 9/4 divided by 6/5
Answer:
45/20
Step-by-step explanation:
Answer:
9/4 divided by 6/5 is 15/8
Step-by-step explanation:
9/4 ÷ 6/5 turns into 9/4 × 5/6 which is the reciprocal
9/4 × 5/6 = 45/24
Divide the numerator and denominator by 3 to get 15/8
15/8 is in its simplest form
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The equation of the parabola is f(x) = 7(x-3)^2+5.
Given:
The equation of the parabola that has the same shape has f(x) = 7x^2 but with vertex(3,5) in the form of f(x) = a(x-h)^2 + k.
If the parabola has same shape then its a values are same.
so f(x) = 7x^2
here a = 7
so the equation with vertex (3,5) also has a value is 7.
(h,k) = (3,5)
f(x) = a(x-h)^2+k
= 7(x-3)^2 + 5.
Therefore the equation of the parabola is f(x) = 7(x-3)^2+5.
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P. USE REPRESENTATIONS Sheila plans to invest $2000 or less in two different accounts. The low
risk account pays 3% annual simple interest, and the high risk account pays 12% annual simple
interest. Sheila wants to make at least $150 in interest this year.
a. Define the variables, then write and graph a system of inequalities to show how Sheila can
split her investment between the accounts.
Inconclusion the amount she invested in each account is $1,000 invested at interest rate 12% .
What is interest rate?
An interest rate informs you of how much borrowing will cost you and how much saving will pay off. Therefore, the interest rate is the amount you pay for borrowing money and is expressed as a percentage of the entire loan amount if you are a borrower.Let x represent the amount invested at 3%.
. 12x +.03(2000-x)= 150
.09x+ 60 = 150
Collect like terms
.09x = 90
Divide both side by .09x
x = 90/.09
x=$1000
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write the quotient as a power. a^21/a
the quotient as a power. a^21/a is a^20.
What is Power?
In mathematics, power defines a base number raised to the exponent, where base number is the factor which is multiplied by itself and exponent denotes the number of times the same base number is multiplied.
What is quotient?
The number we obtain when we divide one number by another is the quotient.
What is exponential function?
An exponential function is a mathematical function of the following form: f ( x ) = a x. where x is a variable, and a is a constant called the base of the function. The most commonly encountered exponential-function base is the transcendental number e , which is equal to approximately 2.71828.
Given that,
a^21/a
quotient as power = a^21/a
= a^(21-1)
= a^20
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Which number is a multiple of 4?
A.) 6
B.) 9
C.) 15
D.) 12
Answer:
D: 12
Step-by-step explanation:
multiples of 4: 4,8,12,16,20
12 is the only number that can be divided by 4 on the list
Hopes this helps please mark brainliest
Answer:
D. 12
Step-by-step explanation:
12 is a multiple of 4 because 4 goes into 12 (evenly, with no remainder).
That's what being a multiple means.
All the "answers" to 4 times table questions are multiples of 4. Here they are:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...
Can't actually list them all because the list goes on forever.
Divide these numbers in the given ratios
36 in the ratio 4:5
Miguel's class just held a class election. 70% of the 30 students in the class voted. How many students voted?
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{70\% of 30}}{\left( \cfrac{70}{100} \right)30}\implies 21[/tex]
Answer: 21
Step-by-step explanation:
70% of 30 students is 21
Hope this helped :)
At the beginning of a chemistry experiment, the volume of liquid in a container was 3 milliliters. During the experiment, the volume dropped to
2.1 milliliters. Find the percent of decrease.
Answer:
30%
Step-by-step explanation:
To calculate the percent of change, you find the amount of change and divide that my the original amount
[tex]\frac{3-2.1}{3}[/tex] = [tex]\frac{.9}{3}[/tex] = .3 As a percent, this would be 30%
im stumped please help
Need help ASAP find the value of x in the triangle below
Answer:
x = 12
Step-by-step explanation:
Creating an equationAngles in a triangle add up to 180 degrees.
This means that 6x - 19 + 3x + 7 + 84 = 180
Solving for x6x - 19 + 3x + 7 + 84 = 180
==> combine like terms
9x + 72 = 180
==> subtract 72 from both sides
9x = 108
==> divide both sides by 9
x = 12
if our circadian clocks are not reset each morning by exposure to light, they: a. tend to run a cycle that is a little longer than 24 hours b. tend to run a cycle that is a little shorter than 24 hours c. tend to run on a 12-hour cycle d. still run exactly on a 24-hour cycle
If our circadian clocks are not reset each morning by exposure to light, they a. tend to run a cycle that is a little longer than 24 hours.
Chronobiology, the study of the body's circadian rhythms, contends that our daily routine, including when we eat, exercise, go to bed, and are exposed to light, may have a significant impact on how our bodies function on a day-to-day basis.
According to the National Institute of General Medical Sciences, circadian rhythms are the normal physical, mental, and behavioral changes that take place in the body on a 24-hour cycle, typically in reaction to light and darkness. One illustration is resting at night and remaining awake during the day.
You might not be able to avoid daytime sleep and evening work if you work an overnight shift or other irregular hours. According to Wright, those people should pay extra close attention to making other healthy decisions a priority, such as getting enough exercise and quitting smoking.
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Write these numbers in standard form.
a) 200000
b) 80000
c) 900 000 000
d) 10 000 000 000
Answer:
a) 2.0 x 10^5
b) 8.0 x 10^4
c) 9 x 10^9
d) 1 x 10^10
Which of the following is(are) the solution(s) to |5x + 2| = 8 ?
Answer:
x = -2 ; x = 6/5
Step-by-step explanation:
5x + 2 >=0
5x >= -2
5/5 x >= -2/5
x >= -2/5
if x >= -2/5
5x +2 = 8
5x = 8 -2
5x = 6
5/5 x = 6/5
x = 6/5
if x < -2/5
-5x -2 = 8
-5x = 8+2
-5x = 10
-5/-5 x = 10/-5
x = -2
10p^2 - 20p - 34 = -4
Answer:
p = -1 ; p= 3
Step-by-step explanation:
10p^2 - 20p - 34 +4 = 0
10p^2 -20p - 30 = 0
10/10 p^2 - 20/10 p - 30/10 = 0
p^2 - 2p - 3 = 0
(p-3)(p+1) = 0
p = 3
p = -1
1. Why is identifying
the domain and range important when
defining a function?
Identifying the domain and range is important when defining a function because domain gives all the input values and range gives all the output values a function can take. It is very essential to know domain and range of a function.
Domain are represented by independent variable and are graphed on x coordinate of the axis.
Range are dependent values and are graphed on y axis of the graph.
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(30 x 5) + (9 x 5) =
150 + 45 = 195
Answer:
thats correct
Step-by-step explanation:
Robin's bank account has $1,876. Every month she withdraws $30 for her phone payment. Create an equation to describe how
much money she will have after a certain amount of months. Write the equation for this problem in function notation. Then
interpret how much money Robin will have after 6 months of payments. Make sure to define your variables and show all of your
work.
Answer: M = 1876 - 30n will give the amount of Money (M) remaining after n months of withdrawing exactly $30 per month, PROVIDED NOTHING ELSE IS EVER ADDED TO OR WITHDRAWN FROM THE ACCOUNT. (Not very realistic.)
Use n=6 to provide the balance after 6 months.
Step-by-step explanation:
(b) Write down the value of 12⁰
(c) Write down the value of 3-²
Answer:
b) 1 c) 0.1Step-by-step explanation:
b) The value of 12⁰ will be?
→ 12⁰ = ?
→ 12⁰ = 1
Hence, the solution is 1.
c) The value of (3)^(-2) will be?
→ (3)^(-2) = ?
→ (3)^(-2) = 0.1
Hence, the solution is 0.1.
Factorise a squared - a -20
Answer:
(a-5)(a+4)
Step-by-step explanation:
[tex]a^{2}[/tex]-a-20
[tex]a^{2}[/tex]+4a-5a-20
a(a+4) -5(a+4)
(a-5)(a+4)
8 The table shows number of bottles of water four
students drank. One bottle holds 335 milliliters. How
many more fluid ounces did Michael drink than
Lydia? (Hint: 1 fl oz ≈ 30 mL)
Kyle = 5 Bottles of Water
Lisa = 4 Bottles of Water
Lydia = 3 Bottles of Water
Michael = 6 Bottles of Water
Michael drinks 33.5 fl oz more fluid ounces than Lydia.
Given in question,
30 ml = 1 fl oz
1 ml = 1/30 fl oz
1 bottle = 335 milliliters
No. of bottles of water Michael drank = 6
= 6 * 335 milliliters
= 2010 milliliters
= 2010/30 fl oz
= 67 fl oz
No. of bottles of water Lydia drank = 3
= 3 * 335 milliliters
= 1005 milliliters
= 1005/30 fl oz
= 33.5 fl oz
Therefore, 67 - 33.5 = 33.5
Hence, Michael drinks 33.5 fl oz more fluid ounces than Lydia.
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if saint cloud is growing according to the equation pn= 200(1.03)n where n is years after 2010, and the population is measured in thousands. find when the population will be 400 thousand.
In 2267 year, this population would be 400000.
From the question, we have
population is growing as Pn = 200*(1.03)n
substituting the value, we get
400000 = 200*(1.03)n
2000 = 1.03n
ln(2000) = n*ln(1.03)
n = ln(20000) / ln(1.03)
n = 257.15
2010 + 257.15 = 2267 year , this population would be 400000.
Multiplication:
Mathematicians use multiplication to calculate the product of two or more numbers. It is a fundamental operation in mathematics that is frequently utilized in everyday life. When we need to combine groups of similar sizes, we utilize multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are multiplied are referred to as the factors. Repeated addition of the same number is made easier by multiplying the numbers.
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help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!