Please help! Daniel incorrectly solved the equations shown. Explain what he did wrong in each solution and then solve both equations correctly.
25x^2-16=9
√(25x^2 )-√16=√9
5x-4=±3
5x=±7
x=±7/5
z^3-2=6
z^3=8
∛(z^3 )=∛8
z=±2
Daniel incorrectly applied the square root to both terms on the left side of the equation and he forgot to take into account the possibility of multiple roots when solving for the cube root of z³.
Let's first look at the first equation that Daniel solved incorrectly:
25x²-16=9
Daniel's solution:
√(25x² )-√16=√9
5x-4=±3
5x=±7
x=±7/5
What Daniel did wrong here is that he incorrectly applied the square root to both terms on the left side of the equation.
Instead, he should have simplified the left side first, using the fact that √(a²) = |a|, before applying the square root. So the correct solution is:
25x²-16=9
25x²=25
x²=1
x=±1
Now let's look at the second equation that Daniel solved incorrectly:
z³-2=6
Daniel's solution:
z³=8
∛(z^3 )=∛8
z=±2
What Daniel did wrong here is that he forgot to take into account the possibility of multiple roots when solving for the cube root of z³.
In fact, z³ = 8 has three roots: z = 2, z = -1 + i√3, and z = -1 - i√3. So the correct solution is:
z³-2=6
z³=8
z=2
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Using the discriminant
Answer:
n = 20/3
Step-by-step explanation:
Theory about the discriminant, and how solutions are acquired via the quadratic formula
One variable quadratic equations in the form [tex]0=ax^2+bx+c[/tex] have exactly one solution when the discriminant [tex](b^2-4ac)=0[/tex].
This is because for quadratic equations of the original form, the two solutions are given by the quadratic formula:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
which is equivalent to [tex]x=\dfrac{-b}{2a}\pm \dfrac{\sqrt{b^2-4ac}}{2a}[/tex].
The [tex]x=\dfrac{-b}{2a}[/tex] is the axis of symmetry, and the [tex]\dfrac{\sqrt{b^2-4ac}}{2a}[/tex] part is how much the solutions deviate from the axis of symmetry. If the deviation is zero, then the two solutions are both ON the axis of symmetry, and are the same number, giving exactly one solution.
Addressing the given equation
Putting it into standard form
Given the given equation...
[tex](n-4)u^2+6=8u[/tex]
subtract 8u from both sides to obtain an equation that looks like the standard form equal to zero...
[tex](n-4)u^2-8u+6=0[/tex]
The variable here is "u" instead of "x", so the solutions to this equation would be [tex]u=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex], where [tex]a=n-4[/tex], [tex]b=-8[/tex], and [tex]c=6[/tex].
Identifying the discriminant
The discriminant then, with values of a, b, and c, substituted, becomes:
[tex]Discriminant=(-8)^2-4(6)(n-4)[/tex]
If we want the discriminant to equal zero (so that there is exactly one solution for "u"), substitute zero on the left side of the equation, and solve for n.
Solving for n
[tex]0=(-8)^2-4(6)(n-4)[/tex]
[tex]0=64-4(6)(n-4)[/tex]
[tex]0=64-24(n-4)[/tex]
Add -24(n-4) to both sides...
[tex]24(n-4)=64[/tex]
Divide both sides by 24...
[tex]n-4=\dfrac{64}{24}[/tex]
Reduce the right hand side...
[tex]n-4=\dfrac{8}{3}[/tex]
Add 4 to both sides...
[tex]n=\dfrac{8}{3}+4[/tex]
Find a common denominator...
[tex]n=\dfrac{8}{3}+\dfrac{12}{3}[/tex]
Find a common denominator...
[tex]n=\dfrac{20}{3}[/tex]
So, if [tex]n=\dfrac{20}{3}[/tex], then the equation [tex](n-4)u^2+6=8u[/tex] has exactly one solution. Furthermore, this is the only value of "n" for which the equation has exactly one solution, because it is the only value of "n" for which the discriminant is zero.
A principal of $3900 is invested at 4.25% interest, compounded annually. How much will the investment be worth after 5 years?
If principal of $3900 is invested at 4.25% interest, compounded annually, the investment will be worth approximately $4,756.89 after 5 years.
To calculate the future value of the investment, we use the formula for compound interest:
A = [tex]P(1 + r/n)^{(n*t)[/tex]
where A is the future value, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
In this case, P = $3900, r = 4.25%, n = 1 (compounded annually), and t = 5. Plugging these values into the formula, we get:
A = 3900(1 + 0.0425/1)⁵
A = 3900(1.0425)⁵
A ≈ $4,756.89
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Find the slope of the line that passes through the given points.
(−1,−1) and (3,−4)
Answer: [tex]\frac{-3}{4}[/tex]
Step-by-step explanation:
To find the slope of the line, we will use change in y over change in x.
[tex]\displaystyle \frac{y_{2} -y_{1} }{x_{2} -x_{1} } =\frac{-4--1}{3--1} =\frac{-4+1}{3+1} =\frac{-3}{4}[/tex]
- x²= +x +12=0 .....................................
Therefore, the solutions of the equation -x² + x + 12 = 0 are x = -3 and x = 6.
What is equation?In mathematics, an equation is a statement that shows the equality between two expressions, typically separated by an equal sign (=). An equation can contain one or more variables, which are symbols that represent unknown or varying values. The value of the variable(s) can be found by solving the equation.
Here,
The given equation is:
-x² + x + 12 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In this case, we have:
a = -1, b = 1, and c = 12
Substituting these values into the quadratic formula, we get:
x= (-1 ± √(1² - 4(-1)(12))) / 2(-1)
x = (-1 ± √(1 + 48)) / (-2)
x = (-1 ± √(49)) / (-2)
x = (-1 ± 7) / (-2)
There are two solutions:
x = (-1 + 7) / (-2)
= -3
x = (-1 - 7) / (-2)
= 6
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22 randomly picked people were asked if they rented or owned their own home, 9 said they rented. Obtain a point estimate of the proportion of home owners. Use a 95% level of confidence.
point
Answer:
True proportion of homeowners in the population lies between 0.333 and 0.847
Step-by-step explanation:
If 9 out of 22 randomly picked people said they rented their homes, then the remaining 13 must own their homes. Therefore, the point estimate of the proportion of homeowners is:
Point estimate = Number of homeowners / Total number of people surveyed
Point estimate = 13 / 22
Point estimate = 0.59 (rounded to two decimal places)
To calculate the 95% confidence interval for the true proportion of homeowners, we can use the following formula:
Confidence interval = Point estimate ± (Z-value x Standard error)
where the Z-value corresponds to the desired level of confidence and the standard error is given by:
Standard error = sqrt [ (Point estimate x (1 - Point estimate)) / Sample size]
For a 95% confidence interval, the Z-value is 1.96 (from the standard normal distribution). Using the point estimate obtained earlier, the sample size is 22, and we can calculate the standard error as:
Standard error = sqrt [ (0.59 x 0.41) / 22 ]
Standard error = 0.131 (rounded to three decimal places)
Substituting these values into the formula, we get:
Confidence interval = 0.59 ± (1.96 x 0.131)
Confidence interval = 0.59 ± 0.257
Confidence interval = (0.333, 0.847)
Therefore, with 95% confidence, we can say that the true proportion of homeowners in the population lies between 0.333 and 0.847.
(2x+1)^2 +(4x-4)^2 = (4x-1)^2
The quadratic equation is solved and the solution is x = 1 and x = 4
Given data ,
Let's expand the left-hand side of the equation:
(2x+1)² + (4x-4)²
= (2x)² + 2(2x)(1) + 1² + (4x)² - 2(4x)(4) + 4² (using the formula for the square of a binomial)
= 4x² + 4x + 1 + 16x² - 32x + 16
= 20x² - 28x + 17
Now let's expand the right-hand side of the equation:
(4x-1)²
= (4x)² - 2(4x)(1) + 1²
= 16x² - 8x + 1
Now we can set the left-hand side equal to the right-hand side and simplify:
20x² - 28x + 17 = 16x² - 8x + 1
4x² - 20x + 16 = 0
x² - 5x + 4 = 0
(x - 1)(x - 4) = 0
Hence , the solution is x = 1 and x = 4
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Workers in a Certain Company are require to pay 5.5% of their salary into a social Security fund. Mr. Mensah has monthly salary of Gld 4500.00 How much Will he pays each month to the Social Security fund.
Mr. Mensah will pay 247.5 each month to the social security fund because 5.5% of 4500 is 247.5.
The total monthly salary of Mr. Mensah is 4500.
He is required to pay 5.5% of his salary into a social security fund.
Now, we have to find the amount he has to pay each month into the social security fund.
To find that, we need to find the value of 5.5 percent of 4500.
4500 ×5.5/100
45×5.5
247.5
Therefore, 5.5% of 4500 is 247.5.
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X 2 3 4 5 6 p(x=x) find the value of p(x>3)
From the given data, we have:
| x | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- |
| p(x) | | | | | |
We need to find the value of p(x > 3).
We know that the sum of all the probabilities is equal to 1. So, we can find the missing probability by subtracting the sum of the probabilities we know from 1.
p(x = 2) + p(x = 3) + p(x = 4) + p(x = 5) + p(x = 6) = 1
We don't know the value of p(x = 2), so we can't directly calculate p(x > 3). But we can find p(x ≤ 3) and subtract it from 1 to get p(x > 3).
p(x ≤ 3) = p(x = 2) + p(x = 3)
To find p(x = 2), we can use the fact that the sum of all the probabilities is 1:
p(x = 2) = 1 - (p(x = 3) + p(x = 4) + p(x = 5) + p(x = 6))
Now we can substitute this into the equation for p(x ≤ 3) and solve:
p(x ≤ 3) = p(x = 2) + p(x = 3)
p(x ≤ 3) = 1 - (p(x = 3) + p(x = 4) + p(x = 5) + p(x = 6)) + p(x = 3)
p(x ≤ 3) = 1 - (p(x = 4) + p(x = 5) + p(x = 6))
Finally, we can subtract p(x ≤ 3) from 1 to get p(x > 3):
p(x > 3) = 1 - p(x ≤ 3)
p(x > 3) = 1 - (1 - (p(x = 4) + p(x = 5) + p(x = 6)))
p(x > 3) = p(x = 4) + p(x = 5) + p(x = 6)
Therefore, the value of p(x > 3) is the sum of the probabilities for x = 4, 5, and 6.
How do l do this please help me asap
Answer:
[tex] {z}^{2} - 10z + 9 = [/tex]
[tex](z - 1)(z - 9)[/tex]
Assertion: 7V5, V2+21 are the irrational number Reason: every integer is a rational number
Assertion: Yes, √2 is an irrational number.
Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.
How to solveBoth (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A).
Assertion: Yes, √2 is an irrational number.
Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.
All real numbers that are not rational numbers are referred to as irrational numbers in mathematics. In other words, it is impossible to describe an irrational number as the ratio of two integers.
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The Complete Question
Assertion (A): V2 is an irrational number. Reason (R) : Decimal expansion of an irrational number is non-recurring and non terminating???? a) Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A) b) Both (A) and ( R) are true but Reason (R) is not a correct explanation of (A) c) Assertion (A) is true and Reason (R) is false d) Assertion (A) is false and Reason (R) is true
Camping A guy rope is attached to the top of a tent pole. The guy rope is pegged into the ground 5 feet from the tent. If the guy rope is 11 feet long, how long is the tent pole?
The Ferris Wheel can carry 4 passengers in each of its 15 cars in one ride. How many passengers can it carry on a total of 35 rides?
The Ferris Wheel can carry 2100 passengers on a total of 35 rides.
According to the question,
Number of cars in the Ferry Wheel = 15
Number of passengers in each car = 4
∴ Number of passengers in 15 cars = 15×4 = 60
So, the number of passengers on the Ferry Wheel = 60
Now,
The number of passengers the Ferry Wheel carries in one ride = 60
∴ The number of passengers in 35 rides = 60×35 = 2100.
Hence the Ferris Wheel can carry 2100 passengers on a total of 35 rides.
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A student wants to compare the amount of money that two local movie theaters make over a two-week period for the last nightly showing of a particular movie. The following box plots show the data for the amount of money each theater makes over the period. Compare the median of each box plot.
Answer:
mt1= 995
mt2=975
Step-by-step explanation:
the line inside the box plot shows where the median is.
HELP FAST!!! EASY ALEGRA 2!
log base 16, 32 can be written as log₂⁴ 2₅.
Evaluate with logₐx, write a and x as powers of same.
a) log₁₆ 64
= log₂⁴=2⁶
b) log₈₁ 27
= log₃⁴ 3³
c) log₁₆ 32
= log₂⁴ 2₅
d) log₂₇ 243
= log₃³ 3₅
Therefore, log base 16, 32 can be written as log₂⁴ 2₅.
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will mark brainliest
Answer:
$74.66
Step-by-step explanation:
Plot the points on the graphing calculator (x = length of HDMI cord, y = cost). Then generate a linear regression model:
y = 3.68x + 1.06 (approximately)
x = 20, so y = 74.66
$74.66 is the correct answer.
fill it in The value of a certain investment over time is given in the table below. Answer the questions below to determine what kind of function would best fit the data, linear or exponential.
Number of Years Since Investment Made, x
1
2
3
4
Value of Investment ($), f(x)
14,944.54
16,357.04
17,974.72
19,891.18
function would best fit the data because as x increases, the y values change
. The
of this function is approximately
.
An exponential function would be more appropriate than a linear function, as exponential growth
the best function for the dataTo determine which type of function best fits the data, we can analyze the difference in the y values (Value of Investment) as x increases.
Differences between consecutive y values:
16,357.04 - 14,944.54 = 1,412.50
17,974.72 - 16,357.04 = 1,617.68
19,891.18 - 17,974.72 = 1,916.46
The differences between consecutive y values are increasing as x increases. This suggests that an exponential function would be more appropriate than a linear function.
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Mannys pay varies directly with the number of lawns he mows. He earns $100 for mowing 4 lawns. Find k in the equation for Mannys pay. Use P=kl
Answer:
10000
otra solución es: 10⁴
Which of the points plotted is farther away from (−7, 8), and what is the distance?
A: Point (5, 8), and it is 11 units away
B: Point (5, 8), and it is 13 units away
C: Point (−7, −5), and it is 12 units away
D: Point (−7, −5), and it is 13 units away
The distance between point (−7, 8) and point (5, 8) is 12 units (since they are on the same horizontal line). The distance between point (−7, 8) and point (−7, −5) is 13 units (using the Pythagorean theorem). Therefore, the point that is farther away is option D: Point (−7, −5), and it is 13 units away.
I need help pleaseeee
(a) The value of the row operation of -2(row1) + row (2) is [-10 4 -12].
(b) The interchange of row 1 and row 2 is [0 6 -3]
[5 -2 6]
What is the row operation of the matrix?The value of the row operation of -2(row1) + row (2) is calculated as follows;
-2 (row 1) = 2 [5 -2 6]
-2 (row 1) = [-10 4 -12]
The addition of -2(row1) to row (2) is calculated as follows;
[-10 4 -12] + [0 6 -3]
= [-10 10 - 15]
The interchange of row 1 and row 2 is calculated as follows;
R₁ ↔ R₂
= [5 -2 6] = [0 6 -3]
[0 6 -3] [5 -2 6]
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The expression 5(2) gives the number of leaves on a plant as a function
of the number of weeks since it was planted.
What does 2 represent in this expression?
Choose 1 answer:
B
The plant was planted 2 weeks ago.
There were initially 2 leaves on the plant.
The number of leaves is multiplied by 2 each week.
C: the number of leaves is multiplied by 2 each week
Step-by-step explanation
The proper expression ought to be 5(2) ^ t =5(2)^{t}
where t speak to the number weeks after it planted.
On the off chance that we put to the time(t) we are going know how numerous clears out at first are. The calculation will be:
5(2)^{t}= 5(2)^{0} =5
This appear that coefficient 5 decide how much the takes off at first are.
Let see how the number going for distinctive esteem of t. On the off chance that its 1 week after plant, the leaf will ended up:
5(2)^{t}= 5(2)^{1} = 10
At that point the number of takes off 2 weeks after planted will be:
5(2)^{t}= 5(2)^{2}= 20
The number keeps twofold each week since coefficient 2 speaks to the proportion of the development of the work and the proportion decides how much takes off duplicated each week.
So, the reply will be C.
A major credit card company is interested in whether there is a linear relationship between its internal rating of a customer’s credit risk and that of an independent rating agency. The company collected a random sample of 200 customers and used the data to test the claim that there is a linear relationship. The following hypotheses were used to test the claim.
H0:β1=0Ha:β1≠0
The test yielded a t
t
-value of 3.34 with a corresponding p
p
-value of 0.001. Which of the following is the correct interpretation of the p
p
-value?
The correct interpretation of the p-value is 3.34.
We are given that;
t-value=3.34
Number of customers= 200
Now,
The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed one, assuming that the null hypothesis is true. In this case, the p-value of 0.001 means that there is a 0.1% chance of getting a t-value of 3.34 or higher (or -3.34 or lower) if the true slope of the linear relationship between the two ratings is zero. The p-value is very small, which indicates that the data provide strong evidence against the null hypothesis and in favor of the alternative hypothesis.
The p-value of 0.001 means that there is a 0.1% chance of getting a t-value of 3.34 or higher (or -3.34 or lower) if there is no linear relationship between the two ratings.
Therefore, by probability the answer will be 3.34
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Can you help me with this question?
Answer:
C
Step-by-step explanation:
It mentions that they charge 1.20 per kg of lettuce
Samantha is designing a new three-dimensional puzzle in the form of a square pyramid. She knows that the design and the general formula for finding the volume are as follows, where h is the height and a is the side length of the base. All of her measurements are in centimeters.
An illustration of a pyramid, with the height of h and width a.
Samantha's puzzle must have a certain volume and height, but she needs to determine the side length of the base to complete the design. Which equation can Samantha use to find the side length of the base in terms of the volume and height of her puzzle?
An equation which Samantha can use to find the side length of the base in terms of the volume and height of her puzzle is: A. a = √(3V/h)
How to calculate the volume of a square pyramid?In Mathematics and Geometry, the volume of a square pyramid can be calculated by using the following formula:
Volume of a square pyramid, V = 1/3 × a² × h
Where:
h represent the height of a square pyramid.a represent the side length of the base of a square pyramid.By making "a" the subject of formula, the side length of the base of a square pyramid can be determined as follows;
V = 1/3 × a² × h
3V = a²h
a² = 3V/h
By taking the square root of both sides of the equation, we have:
a = √(3V/h)
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Help please
Condense to a single logarithm is possible
In(6x^9)-In(x^2)
The logarithm expression can be simplified to:
In(6x^9)-In(x^2) =7·ln(6x)
How to write this as a single logarithm?There are some logarithm properties we can use here.
log(a) - log(b) = log(a/b)
log(a^n) = n*log(a)
(these obviously also apply to the natural logarithm)
Now let's look at our expression, it says that:
In(6x^9)-In(x^2)
Using the first rule, we can rewrite this as.
In(6x^9)-In(x^2) = ln(6x^9/x^2)
Now solving the quotient in the argument:
ln(6x^9/x^2) = ln(6x^7) = 7·ln(6x)
That is the expresison simplified.
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If you look at these six numbers, you can see they range from 98 all the way to 642. Would the best interval to use for this group be 10 or 100?
If you look at the six numbers (101, 203, 407, 512, 98, 642) you can see they range from 98 all the way to 642. the best interval to use for this group be 100.
What is the range?The right interval to utilize for this set of numbers depends on the aim of the analysis.
If we want to look on the single values and small differences between them, then using interval of 10 would be more better. This would give u room to see small changes that exist between adjacent numbers.
So, in the event that we need to look on the full range and distribution of the numbers, an interval of 100 would be more better. This would permit us to see how the numbers are shared over a larger area of values.
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See full question below
If you look at these six numbers, you can see they range from 98 all the way to 642. Would the best interval to use for this group be 10 or 100?
101, 203, 407, 512, 98, 642
In 2015, there were roughly 1 x 10° high school football players and 2x 10° professional football players in the United States. About how many times more high school football players were there?
The number of times more high school football players were there is 500.
Given that, in 2015, there were roughly 1×10⁶ high school football players and 2×10³ professional football players in the United States.
Here, the number of times more high school football players were there
= 1×10⁶/2×10³
= 1×10³/2
= 1000/2
= 500
Therefore, the number of times more high school football players were there is 500.
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"Your question is incomplete, probably the complete question/missing part is:"
In 2015, there were roughly 1×10⁶ high school football players and 2×10³ professional football players in the United States. About how many times more high school football players were there?
The bells obtain a 30-year, $90,000 conventional mortgage at a 10.0% rate on a house selling for $120,000. Their monthly mortgage payment, including principal and interest, is $783.00. They also pay 2 points at closing.
A)What is total amount the Bells will pay for their house.
B) How much of the cost will be interest (including the 2 points)?
C) How much of the first payment on the mortgage is applied to the principal?
The total amount that the Bells will pay for their house would be $313, 680.
The cost that would be interest is $193, 680
The amount of the first payment going to mortgages is $165. 24
How to find the total amount paid ?Find the down payment :
Down payment = House price - Mortgage amount = $120,000 - $90,000 = $30,000
Then the points paid at closing :
= $ 90,000 x 0. 02 = $1,800
Total amount paid = Total mortgage payments + Down payment + Points cost
= 281, 880 + 30, 000 + 1, 800
= $ 313, 680
The total that is interest is:
Total interest paid = Total interest paid through monthly payments + Points cost
Total interest paid = $ 191, 880 + $ 1, 800 = $193, 680
The amount of the first payment going to principal is:
Principal portion of first payment = 90, 000 - 89, 834.76 = $165. 24
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Select the reason that best supports Statement 5 in the given proof.
Where is the given proof
Can someone help me with this pls
Answer:
Subtract the powers, 8-4.
When you simplify that equation itself, the answer is 10,000.
So when you do 10⁴, it's 10,000.
