[tex]x^4-625=(x^2-25)(x^2+25)=(x-5)(x+5)(x-5i)(x+5i)\\\\x^4+10x^2+25=(x^2+5)^2=((x-\sqrt5i)(x+\sqrt5i))^2=(x-\sqrt5i)^2(x+\sqrt5i)^2[/tex]
Answer:
1) [tex]x^4-625=(x+5i)(x-5i)(x+5)(x-5)[/tex]
2) [tex]x^4+10x^2+25=(x+\sqrt{5}i)^2(x-\sqrt5 i)^2[/tex]
Step-by-step explanation:
1) Given : Expression [tex]x^4-625[/tex]
To find : Factor the expression completely over the complex numbers ?
Solution :
We can re-write the expression as,
[tex]x^4-625=(x^2)^2-(25)^2[/tex]
Applying identity, [tex]a^2-b^2 = (a+b)(a-b)[/tex]
[tex]x^4-625=(x^2+25)(x^2-25)[/tex]
[tex]x^4-625=(x^2+25)(x^2-5^2)[/tex]
Again apply same identity,
[tex]x^4-625=(x^2+25)(x+5)(x-5) [/tex]
The factor of [tex]x^2+25=(x+5i)(x-5i)[/tex]
Factor form is [tex]x^4-625=(x+5i)(x-5i)(x+5)(x-5)[/tex]
2) Given : Expression [tex]x^4+10x^2+25[/tex]
To find : Factor the expression completely over the complex numbers ?
Solution :
Expression [tex]x^4+10x^2+25[/tex]
Let [tex]x^2=y[/tex]
[tex]y^2+10y+25[/tex]
To factor we equate it to zero.
[tex]y^2+10y+25=0[/tex]
Apply middle term split,
[tex]y^2+5y+5y+25=0[/tex]
[tex]y(y+5)+5(y+5)=0[/tex]
[tex](y+5)(y+5)=0[/tex]
Substitute back,
[tex](x^2+5)(x^2+5)=0[/tex]
[tex](x^2+5)^2=0[/tex]
[tex]x^2+5=0[/tex]
[tex]x^2=-5[/tex]
Taking root both side,
[tex]x=\sqrt{-5}[/tex]
[tex]x=\pm \sqrt{5}i[/tex]
So, The factors are [tex](x+\sqrt{5}i)(x-\sqrt5 i)[/tex]
Factor form is [tex]x^4+10x^2+25=(x+\sqrt{5}i)^2(x-\sqrt5 i)^2[/tex]
can someone tell me how to evalaute 6!
Where is the question?
If you're referring to factorials, then,
6! = 6*5*4*3*2*1 = 720
You start with 6 and count your way down to 1, multiplying along the way.
Amy invested $223 in the bank and a year later has $280.98. By what percent has the amount changed?
Answer:
26% increase
Step-by-step explanation:
Answer:
26% increase
Step-by-step explanation:
A line goes through the points (-6, -8) and (12, 7). a) What is the slope of the line? Show your work. b) Write the equation of the line in point-slope form. Show your work c) Write the equation of the line in slope-intercept form. Show your work. Answer: a) b) c)
m = [tex]\frac{5}{6}[/tex] , y - 7 = [tex]\frac{5}{6}[/tex] ( x - 12), y = [tex]\frac{5}{6}[/tex] x - 3
(a) calculate the slope using the gradient formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (- 6, - 8) and (x₂, y₂) = (12, 7)
m = [tex]\frac{7+8}{12+6}[/tex] = [tex]\frac{15}{18}[/tex] = [tex]\frac{5}{6}[/tex]
(b) the equation of a line in point-slope form is
y - b = m(x - a )
where m is the slope and (a, b) a point on the line
using m = [tex]\frac{5}{6}[/tex] and (a , b) = (12, 7 )
y - 7 = [tex]\frac{5}{6}[/tex] (x - 12)
(c) the equation of a line in slope-intercept form is
y = mx + c ( m is the slope and c the y-intercept )
rearrange (b ) into this form
y - 7 = [tex]\frac{5}{6}[/tex] - 10
y = [tex]\frac{5}{6}[/tex] x - 3
Multiply the polynomials. (4x^2+3+7)(8x-5)
The point-slope form of the equation of the line that passes through (–5, –1) and (10, –7) is y + 7 = (x – 10). What is the standard form of the equation for this line?
2x – 5y = –15
2x – 5y = –17
2x + 5y = –15
2x + 5y = –17
Answer: The correct option is (C) [tex]2x+5y=-15.[/tex]
Step-by-step explanation: Given that the equation of the line that passes through (-5, -1) and (10, -7) in point-slope form is given by
[tex]y+7=-\dfrac{2}{5}(x-10)~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to find the standard form of the equation for the above line.
We know that the STANDARD form of the equation of a line is given by
[tex]ax+by=c,~~~~~\textup{[a and b cannot be zero at the same time]}.[/tex]
From equation (i), we have
[tex]y+7=-\dfrac{2}{5}(x-10)\\\\\Rightarrow 5(y+7)=-2(x-10)\\\\\Rightarrow 5y+35=-2x+20\\\\\Rightarrow 2x+5y=20-35\\\\\Rightarrow 2x+5y=-15.[/tex]
Thus, the required standard form is [tex]2x+5y=-15.[/tex]
Option (C) is CORRECT.
Use the exponential regression equation that best fits the data
(10,4), (12,20), (12,20), (13,35), and (16,300), to estimate the value of y when x = 14.
A. 48.4
B. 64.9
C. 132.3
D. 223.7
To estimate the value of y when x = 14 using exponential regression, we first find the equation that best fits the data, and then substitute x = 14 into it. The estimated value of y when x = 14 is approximately 223.7.
Explanation:To find the value of y when x = 14 using exponential regression, we need to first find the equation that best fits the data. We can do this by using a graphing calculator or software that can perform exponential regression. Once we have the equation, we can substitute x = 14 into it to find the estimated value of y.
The exponential regression equation that best fits the data is [tex]y = 4 \times 2.0487^x.[/tex] Plugging in x = 14 into this equation gives us [tex]y = 4 \times 2.0487^{14} = 223.7.[/tex]
Therefore, the estimated value of y when x = 14 is approximately 223.7.
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To estimate the value of y when x = 14 using exponential regression, we first need to find the exponential equation that best fits the given data points. The estimated value of y when x = 14 is approximately D. 223.7.
Explanation:To estimate the value of y when x = 14 using exponential regression, we first need to find the exponential equation that best fits the given data points.
We can do this by using a graphing calculator or a statistical software. Once we have the equation, we can substitute x = 14 into the equation to find the estimated value of y.
Using the given data points, the exponential regression equation that best fits the data is y = 3.6841 * (1.5693)^x. Substituting x = 14 into the equation, we get: y = 3.6841 * (1.5693)^14 ≈ 223.7.
Therefore, the estimated value of y when x = 14 is approximately 223.7. Therefore, the correct answer is D. 223.7.
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When mindy went to china she exchanged her $1 for 6.589 yaun. What place is in the hundredths place of 6.589?
The 8 is in the hundredths place.
Hope this helps & good luck. :)
Bobbi invests the money he has saved in an investment fund at the bank. The fund pays him interest each month according to the following sequence $0,$10,$20,$30,… meaning he received $0 in interest at month 0, $10 in interest after the first month, $20 in interest after the second month, and so on.
If f(n) represents the sequence, determine the amount of interest he will receive after 12 months.
Enter the value of f(n) at f(12).
$ [blank] −−−−−−
The value of f(12) which represents the amount of interest he will receive after 12 months is; $120.
Evidently, the sequence is an arithmetic progression;
As such, we must evaluate the common difference, d as follows;
d = f(2) - f(1)d = 20 - 10d = $10.The first term, a = $10
Therefore; From the nth term formula for an arithmetic progression;
f(12) = $10 + 11 ($10)f(12) = $10 + $110f(12) = $120Read more:
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Dan buys 24 packets of nuts. Each packet of nuts weighs 225g. Work out the total weight of all the packets of nuts that dan buys.
Answer:
Total weight of the nuts bought by Dan = 5.4 kg
Explanation:
Number of packets of nuts bought by Dan = 24
Weight of each packet of nuts = 225g
Total weight of the nuts bought by Dan = Number of packets of nuts bought by Dan * Weight of each packet of nuts
Total weight of the nuts bought by Dan = 24 * 225 = 5400 g = 5400/1000 kg = 5.4 kg
Total weight of the nuts bought by Dan = 5.4 kg
Find the maximum value of C=3x+4y
Subject to the following constraints
x≥2
x≤5
y≥1
x≤6
if x can be 2-5 and y can be 1-6, c=3(5)+4(6)
which would turn out to be c=39
The maximum value of C will be equal to 39.
What is an equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
Given constraints are:-
x ≥ 2
x ≤ 5
y ≥ 1
y ≤ 6
Here maximum value of x = 5 and for y = 6
Putting the maximum values of x and y to calculate the maximum value of C.
C = 3x + 4y
C = ( 3 x 5 ) + ( 4 x 6 )
C = 15 + 24
C = 39
Therefore the maximum value of C will be equal to 39.
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The side length, s, of a cube is 3x + 2y. If V = s3, what is the volume of the cube? 3x3 + 18x2y + 36xy2 + 8y3 27x3 + 54x2y + 18xy2 + 2y3 27x3 + 18x2y + 12xy2 + 2y3 27x3 + 54x2y + 36xy2 + 8y3
We are given side length of a cube = (3x+2y).
Volume is given by formula [tex]V=s^3[/tex], where s is the side of cube.
We have s = (3x+2y).
Plugging s = (3x+2y) in formula now.
We get,
[tex]V=(3x+2y)^3[/tex]
Expanding by applying formula [tex](a+b)^3=(a)^3 + 3a^2b+3b^2a+(b)^3.[/tex]
[tex](3x+2y)^3 =(3x)^3+3(3x)^2(2y)+3(2y)^2(3x)+(2y)^3[/tex]
[tex](3x+2y)^3 = 27x^3+54x^2y+36xy^2+8y^3[/tex]
Therefore, correct option is 4th option : [tex]27x^3 + 54x^2y + 36xy^2 + 8y^3[/tex].Answer:
D
The side length, s, of a cube is 3x + 2y. If V = s3, what is the volume of the cube?
3x3 + 18x2y + 36xy2 + 8y3
27x3 + 54x2y + 18xy2 + 2y3
27x3 + 18x2y + 12xy2 + 2y3
27x3 + 54x2y + 36xy2 + 8y3
Which system of linear inequalities is graphed? {y>2x+1x+y<−2 {y<2x+1x+y>−2 {y≤2x+1x+y≥−2 {y≥2x+1x+y≤−2 The image is a system of linear inequalities graphed on a coordinate plane with increments of 1 and x and y axis ranging from negative 5 to 5. A dashed line passes through the points begin ordered pair 0 comma 1 end ordered pair and begin ordered pair 1 comma three end ordered pair. The shading is above the line. The other line passes through begin ordered pair 0 comma negative 2 end ordered pair and begin ordered pair negative 2 comma zero end ordered pair. The shading is below this line.
Let us check the slope of y-intercepts of the given lines in the graph first.
The y-intercept of above line is 1 and slope of
Rise/run = 2/1 (Moving 2 units up and 1 unit right).
So, the equation should be
y=2x+1
Y-intercept of second line is -2 and slope is
Rise/run = -1/1 (Moving 1 unit down and 1 unit right).
So, the equation should be
y=-x-2.
Now, we need to check the shaded portion for inequality signs.
We have both lines dotted.
So, the inequality signs would be just < or >.
For y=2x+1 line : Shading is on the left side.
On the left side of the line y=2x+1, the y-values are greater than on right side.
Therefore, we got first inequality y>2x+1
For y=-x-2 line : Shading is on the down of the line.
On the down of the line y=-x-2, the y-values are less than up side of the line.
Therefore, second inequality would be
y<-x-2 or y+x<-2
Therfore, correct option is first option y>2x+1 and y+x<-2.
Answer:
A was the answer.
Step-by-step explanation:
i took the test.
I do not know how to do this need help
Remark
First of all you have to declare the meaning of g(f(x)) After you have done that, you have to make the correct substitution.
Givens
f(x) = 4x^2 + x + 1
g(x) = x^2 - 2
Discussion
What the given condition g(f(x)) means is that you begin with g(x). Write down g(x) = x^2 - 2
Wherever you see an x on either the left or right side of the equation, you put fix)
Then wherever you see f(x) on the right you put in what f(x) is equal to.
Solution
g(x) = x^2 - 2
g(f(x)) = (f(x))^2 - 2
g(f(x)) = [4x^2 + x + 1]^2 - 2
f(x)^2 =
4x^2 + x + 1
4x^2 + x + 1
16x^4 + 4x^3 + 4x^2
4x^3 + x^2 + x
4x^2 + x + 1
16x^4 + 8x^3 + 9x^2 + 2x + 1
Answer
g(f(x)) = 16x^4 + 8x^3 + 9x^2 + 2x + 1 - 2
g(f(x)) = 16x^4 + 8x^3 + 9x^2 + 2x - 1
Solve for a.
5 + 14a = 9a - 5
Since you are solving for a, you want to have a on one side of the equation and the other terms on another side of the equation. It would be easiest to have all the terms with a on the left side of the equation, so that is what we will do.
Subtract 9a from both sides to get a on the left side of the equation.
5 + 5a = -5
Subtract 5 from both sides of the equation to isolate the term with a.
5a = -10
Divide both sides of the equation by 5 to solve for a.
a = -2
The result to the equation is a = -2.
To break for a in the equation 5 + 14a = 9a - 5, we can start by simplifying both sides of the equation.
Adding 5 to both sides
5 + 14a + 5 = 9a - 5 + 5
10 + 14a = 9a
Next, we can insulate the variable a by abating 9a from both sides
14a - 9a + 10 = 9a - 9a
5a + 10 = 0
To break for a, we'll abate 10 from both sides
5a + 10 - 10 = 0 - 10
5a = -10
Eventually, we divide both sides by 5 to break for a
5a/ 5 = -10/ 5
a = -2
Thus, the result to the equation is a = -2.
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) Given the conditional statement, "If an integer is a counting number, then it is 0," which of the following can be concluded?
The statement is false because the hypothesis is false.
The statement is false because the conclusion is false.
The statement is false because both the hypothesis and the conclusion are false. The statement is true.
Answer:
The statement is false because the conclusion is false.
Step-by-step explanation:
"then it is 0" is the conclusion. This is false because not all integers are 0.
Fia's goal was to raise $5,000 to help protect the manatees. She raised $3,500. What percent of her goal did she achieve
In order to turn a ratio into a percentage, you have to turn the given fraction into an equivalent fraction with denominator 100. So, you have
[tex] \dfrac{3500}{5000}=\dfrac{x}{100} [/tex]
Solving for x, you get
[tex] x=\dfrac{3500\cdot 100}{5000} = \dfrac{3500}{50} = \dfrac{350}{5} = 70 [/tex]
So, they achieved the 70% of their goal.
Tim can complete 168 math problems in six minutes. How many problems could he complete in one minute?
For each charity donation made by an employee of Tillman Corporation, the company will contribute an additional 40% of the amount. Last month, Tillman donated $3540. If 118 employees donated, what was the average contribution of each employee?
A) $57
B) $60
C) $75
D) $80
E) $90
Charity donated by Tillman = $3540
Number of employees who donated = 118
Contribution of Tillman = 40%
Let the average contribution of employees be = x
Then the amount contributed by employees = 118x
Then contribution of Tillman = [tex]\frac{40}{100}\times118x[/tex]
As Tillman donated 3540 , then
[tex]3540=\frac{40}{100}\times118x[/tex]
[tex]3540=47.20x[/tex]
[tex]x=75[/tex]
Hence, average contribution by each employee is $75.
Option C is correct.
Answer:
$75
Tillman's donation = .40(total employees' donations)
$3540 = .40x
x = 8850
8850
118
= $75 for each employeeep explanation:
$75
Tillman's donation = .40(total employees' donations)
$3540 = .40x
x = 8850
8850
118
= $75 for each employee
Luke can paint 91 portraits in 7 weeks. How man portraits can luke paint in 4 weeks
91/7= 13
13 portraits in 1 week
13x4=52
52 portraits in 4 weeks
the question is in the picture
Remark
Usually this is written in very simple terms. You could do it as a proportion, but it might be easier to see if you used Newton's formula twice.
Formula
F = m * a
m = mass, F = force, a = accelerationn
Givens
F1 = 70 N
a1 = 6.5 m/s^2
m = ?
F2 = ??
m = ?
a2 = 8 m/s^2
Step One
Solve for m
F1 = m * a1 Substitute the values
70 = m * 6.5 Divide by 6.5
70/6.5 = m
10.77 kg = m
Step Two
Solve for F2
m = 10.77 kg
a = 8
F2 = ??
F2 = m * a2
F2 = 10.77 * 8
F2 = 86.154
Answer: C rounded to the nearest 1/100
If f(x) varies directly with x2 and f(x)=10 what is the value of f(x) when x=3
If f(x) varies directly with [tex]x^2[/tex]
If f(x) varies directly with x then we use equation f(x) = kx
where k is the constant of proportionality
So equation becomes [tex]f(x) = kx^2[/tex]
We use the information and find out k
f(x)= 10
[tex]f(x) = kx^2[/tex]
[tex]10= kx^2[/tex]
[tex]k = \frac{10}{x^2}[/tex]
now we use the value of k and find the value of f(x) when x=3
[tex]f(x) = \frac{10}{x^2}*x^2[/tex]
f(x) = 10
The value of f(x) = 10 when x= 3
The value of f(x) when x=3 is 90.
If f(x) varies directly with x2 and is given that f(x)=10 when x=1 (since 1 squared is 1), we can express the direct variation as f(x) = k × x2, where k is the constant of variation. To find the value of k, we use the information that f(1)=10. Thus, 10 = k × 12, which means k=10.
Now that we know k=10, we can find f(x) when x=3. We substitute x with 3 in the equation f(x) = 10 × x2, yielding f(3) = 10 × 32 = 10 × 9 = 90. Therefore, the value of f(x) when x=3 is 90.
Nadia and Reggie want to rent bikes with two other friends. They have $150 to spend on bike rentals. The sign below show the bike rental rates.
Based on the information on the sign, the equation below can be used to determine the number of hours, h , the 4 friends can rent bikes with $150.
4 (9.75h - 3) = 150
Nadia says they have enough money to rent bikes for a maximum of 3 hours. Is Nadia correct? Show all of your work.
No! Nadia is not correct, the maximum is actually 4 hours. I know this because
4(9.75h+3)=150 distribute
39h-12=150 get the value h alone by adding 12
39h = 162 divide both sides by 39
h=4.15
No, Nadia is incorrect, the maximum is actually 4 hours.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
We are given that the equation below can be used to determine the number of hours, h , the 4 friends can rent bikes with $150.
4 (9.75h - 3) = 150
Solving;
4(9.75h+3)=150
Now distribute
39h-12=150
To get the value h alone by adding 12;
39h = 162
Then divide both sides by 39
h=4.15
She have enough money to rent bikes for a maximum of 4 hours.
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Where does 9.34612219 go on the tenths place?
when rounded to the tenths place it would be 9.3
If the parent function is y = 1/x, describe the change in the equation y = 1/(x+4)
a) Moves 4 units to the left
b) Moves 4 units to the right
c) Moves 4 units up
d) Moves 4 units down
f(x) + n - shift a graph n units up
f(x) - n - shift a graph n units down
f(x + n) - shift a graph n units left
f(x - n) - shift a graph n units right
-------------------------------------
We have:
[tex]y=\dfrac{1}{x}\to y=\dfrac{1}{x+4}\\\\f(x)=\dfrac{1}{x}\to f(x+4)=\dfrac{1}{x+4}[/tex]
Answer: a) Moves 4 units to the leftFind the solutions of the given equation.
25x^2+64=289
Answer:
D
Step-by-step explanation:
Start by subtracting 64 from both sides:
[tex]25x^2=225[/tex]
Factor this by taking roots. Divide both sides by 25 first:
[tex]x^2=9[/tex]
We "undo" a square by taking the square roots of both sides. Taking the square root leaves the possibilities of both the positive and negative roots of 9. Therefore, the solutions to this are
[tex]x=\sqrt{3},-\sqrt{3}[/tex]
Which expression are polynomial can be more than one answer
Answer:
The first and third one
Step-by-step explanation:
Answer:
[tex]\frac{7y^{2} +9x}{4}[/tex] and 6y²+5x
Step-by-step explanation:
Polynomials are algebraic expressions involving one, or even more than one variables raised to any power summing each other. Generally speaking, in other words an one variable Polynomial is
[tex]a_{0}+a_{1} x+a_{2} x^{2}...a_{n} x^{n}[/tex]
So a multivariate polynomial expression are [tex]\frac{7y^{2} +9x}{4}[/tex] and 6y²+ 5x.
Because in the other options:
We have a Radical of an Irrational number[tex]\sqrt{3} =3^{\frac{1}{3}}[/tex] times x (2nd option) what does not configure a polynomial.
And a Polynomial Quotient with a variable on the denominator, described by:
[tex]P(x)=\frac{Q(x)}{R(x)} \\ P(x)=q(x)*Q(x)+R(x)[/tex]
Like the last option:
[tex]\frac{7x^{2}+8}{4x}[/tex]
Suppose that the functions r and s are defined for all real numbers x as follows.
r(x) = x-1
s(x) = 3x^2
Write the expressions for (r+s) (x) and (r*s) (x) and evaluate (r-s) (-3).
(r+s) (x) =
(r*s) (x) =
(r-s) (-3) =
Please help.
If 2 lemons cost 15cent how many can be bought for 60cent
You could buy 8 lemons for 60 cents.
Is the function even odd or neither?
The answer is the first option: Even.
The explanation for this exercise is shown below:
1. By definition, if [tex]f(x)=f(-x)[/tex] the fucntion is even.
2. When the graph is symmetric with respect to the y-axis, it is an even function.
3. As you you can see in the graph attached in the problem, the graph is symmetric about the y-axis. Therefore, you can conclude it is an even function.
An alloy consists of nickel, zinc, and copper in the ratio 2:7:9. How many pounds of nickel have to be used to create alloy that contains 4.9 lb of zinc?
Final answer:
To create an alloy with 4.9 lb of zinc based on the nickel to zinc ratio of 2:7, about 1.4 pounds of nickel is required.
Explanation:
To determine how many pounds of nickel have to be used to create an alloy that contains 4.9 lb of zinc, we must first understand the given ratio of nickel to zinc to copper in the alloy, which is 2:7:9. Since zinc's ratio is 7 and we have 4.9 lb of it, we can calculate the amount of nickel by setting up a proportion.
We use the proportion 2/7 = x/4.9, where x represents the pounds of nickel. By cross-multiplying and solving for x, we get x = (2 * 4.9) / 7, which simplifies to x = 9.8 / 7. This gives us the result x ≈ 1.4 lb.
Therefore, to create an alloy with 4.9 lb of zinc, approximately 1.4 pounds of nickel are needed.