The probability that [tex]\(x^2 + 7x + k\)[/tex] is factorable for [tex]\(0 \leq k \leq 20\) and \(k\)[/tex] is an integer is [tex]\(\frac{7}{21}\)[/tex].
The discriminant of [tex]\(x^2 + 7x + k\)[/tex] is [tex]\(7^2 - 4(1)(k) = 49 - 4k\)[/tex].
For the quadratic to be factorable, [tex]\(49 - 4k\)[/tex] must be a perfect square. Let's denote [tex]\(49 - 4k\)[/tex] as [tex]\(m^2\)[/tex], where m is an integer. Then we have: [tex]\[49 - 4k = m^2\][/tex]
Rearranging the equation gives us:
[tex]\[4k = 49 - m^2\][/tex]
[tex]\[k = \frac{49 - m^2}{4}\][/tex]
Since k must be an integer, [tex]\(49 - m^2\)[/tex] must be divisible by 4. This means that [tex]\(m^2\)[/tex] must be of the form [tex]\(4n + 1\) or \(4n - 1\)[/tex] because 49 is 1 more than a multiple of 4 (since 48 is a multiple of 4).
We are looking for values of m such that [tex]\(m^2\)[/tex] is less than or equal to 49 (since [tex]\(k \geq 0\)),[/tex] and [tex]\(m^2\)[/tex] is of the form [tex](4n + 1\) or \(4n - 1\).[/tex] The values of m that satisfy this are [tex]\(m = \pm1, \pm3, \pm5, \pm7\)[/tex].
For each of these values of m, we can calculate the corresponding value of k:
For [tex]\(m = 1\): \(k = \frac{49 - 1^2}{4} = 12\)[/tex]
For [tex]\(m = 3\): \(k = \frac{49 - 3^2}{4} = 11\)[/tex]
For [tex]\(m = 5\): \(k = \frac{49 - 5^2}{4} = 6\)[/tex]
For [tex]\(m = 7\): \(k = \frac{49 - 7^2}{4} = -4\)[/tex] (not within the range [tex]\(0 \leq k \leq 20\)[/tex])
For negative values of m, we get the same values of k as for the positive values, so we don't need to consider them separately.
Thus, the values of k that make the quadratic factorable are 6, 11, and 12. There are 21 possible values for k in the range [tex]\(0\) to \(20\)[/tex] inclusive. Therefore, the probability that [tex]\(x^2 + 7x + k\)[/tex] is factorable is the number of favourable outcomes divided by the total number of possible outcomes: [tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{3}{21} = \frac{1}{7}\][/tex]
Upon re-evaluating the calculation, we realize that the value of k corresponding to m = 7 is not valid since it results in a negative number, which is outside our specified range for k. Therefore, we should not count it. The correct count of valid k values is 3 (for m = 1, 3, 5, and since each of these m values corresponds to two k values (one for m) and one for -m, we have 6 valid k values in total.
Given that there are 21 possible integer values for k in the range 0 to 20 inclusive, the correct probability is:
[tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{6}{21} = \frac{2}{7}\][/tex]
However, this contradicts the initial statement that the correct answer is [tex]\(\frac{7}{21}\)[/tex]. We must correct this to ensure the final answer is accurate.
The correct probability is obtained by considering that for each valid m (excluding [tex]\(m = 0\)),[/tex] there are two corresponding values of k (one for m and one for -m. Since we have valid m values of 1, 3, 5, and their negatives, we have a total of 6 valid k values. But since m = 7 and m = -7 are not valid (as they result in k = -4), we should not include them. Therefore, the correct number of valid k values is 6, and the correct probability is: [tex]\[\frac{\text{Number of factorable } k}{\text{Total number of } k} = \frac{6}{21}\][/tex]
This simplifies to: [tex]\[\frac{6}{21} = \frac{2}{7}\][/tex]
Thus, the correct probability is [tex]\(\frac{2}{7}\), not \(\frac{7}{21}\)[/tex]. The initial statement contained an inaccuracy, and the correct answer is [tex]\(\frac{2}{7}\)[/tex].
a photograph measures 10 inches by 12 inches. a frame with constant width is placed around the photograph so that the total area of the framed photo is 288 square inches. what is the width of the frame? enter your answer in the box.
Answer:
3
Step-by-step explanation:
took test rn i ithink they ar right
Miguel has $20. He spends $6.25 on a movie ticket, $3.70 for snacks, and $1.25 for bus fare each way.
Miguel wants to rent a boat and spend at most $52. The boat costs $7 per hour, and Miguel has a discount coupon for $4 off. What are the possible numbers of hours Miguel could rent the boat?
Use t for the number of hours- write your answer as an inequality solved for t. this is due today plz help thank you!
Hans needs to rent a truck. suppose company A charges a rate of $40 per day and company B charges a $60 fee plus $20 per dat. for what number of days is the cost of the same
If there are 14 people sitting evenly spaced around a circle which person is directly across from the 2nd person?
Answer:
The 9th
Equation:
14/2=7 so 7 and 14 facing each other
Draw a circle with that
Draw 6 dots on each blank side
2 rom 14 is 2 so 7+2=9
So the answer is 9th
To determine which person is directly across from the 2nd person in a circle with 14 people, person 9 is directly across from the 2nd person.
Explanation:To determine which person is directly across from the 2nd person in a circle with 14 people, we need to use modular arithmetic. If we number the people in the circle from 1 to 14, the person directly across from the 2nd person would be 9th. This can be found by adding 7 (the total number of people in the circle divided by 2) to 2, and then taking the remainder when divided by 14. Therefore, person 9 is directly across from the 2nd person.
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A 25-N net force is applied to a rolling cart and produces an acceleration of 5 m/s^2. What is the cart's mass?
Answer:
The cat's mass would be 5 kg.
Which correlation coefficient indicates the stronger linear relationship between random variables for a fixed sample size?
A. r= 0.8
B. r= 0.5
C. r= -0.2
D. r= -0.9
The required among the given options, the correlation coefficient with the strongest linear relationship is r = 0.8.
What is the correlation coefficient indicates?The correlation coefficient is a measure of the strength and direction of the linear relationship between two random variables. The value of the correlation coefficient ranges from -1 to +1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and +1 indicates a perfect positive linear relationship.
Among the given options, the correlation coefficient with the strongest linear relationship is r = 0.8. This is because the closer the value of the correlation coefficient is to +1 or -1, the stronger the linear relationship between the two random variables. A correlation coefficient of 0.5 indicates a moderate positive relationship, a correlation coefficient of -0.2 indicates a weak negative relationship, and a correlation coefficient of -0.9 indicates a strong negative relationship.
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How many ways can a jury of four men and three women be selected from twelve men and ten women
Final answer:
The number of ways to choose the jury is 59,400.
Explanation:
The question is concerned with a combinatorial problem of selecting a jury from a larger pool of people, which is a classic example of the use of combinations in probability and statistics.
To find out how many ways we can select a jury of four men from twelve men, and three women from ten women, we use the combination formula which is C(n, k) = n! / (k! * (n-k)!), where '!' represents factorial, 'n' is the total number of people, and 'k' is the number of people to choose.
To select the men, we calculate the combinations as follows:
C(12, 4) = 12! / (4! * (12-4)!) = 495 waysTo select the women, we calculate the combinations as follows:
C(10, 3) = 10! / (3! * (10-3)!) = 120 waysTo find the total number of ways to select the jury of four men and three women, we multiply the number of ways to choose the men by the number of ways to choose the women:
495 ways (to choose men) * 120 ways (to choose women) = 59,400 different ways to select the jury.
QUICK AND URGENT
Marvin can run 48km in 6 hours.
How many kilometers can he run in 6hours 30min
And how do you do it.
Graph the linear equation find three points that solve the equation then plot on the graph x-3y=-6 plz help
The linear equation x - 3y = -6 can be rearranged as y = (1/3)x + 2. By substituting x-values of 0, 3, and 6, we obtained three points (0,2), (3,3), and (6,4). Plotting these points to graph helps us to visualize the equation.
Explanation:The original linear equation is x - 3y = -6. We can rearrange this equation to y = (1/3)x + 2. From this, we can derive three points on the graph. For example, let's calculate the y-values for x = 0, 3, and 6, which are easy to use.
For x = 0, the equation becomes y = (1/3)(0) + 2 = 2. So, the point (0,2) lies on this line.
For x = 3, the equation becomes y = (1/3)(3) + 2 = 3. So, the point (3,3) is on the line. For x = 6, the y-value is y = (1/3)(6) + 2 = 4, giving us the point (6,4).
These three points can be plotted on a graph and a line can be drawn through them to graph the equation.
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On a coordinate grid, the distance between point F and point G is 13 units. Point F is located at (-1, 3). Point G is located at (x, -9). Which value could be the value of x?
Which of the fallowing are proportions?
Simplify 5 − (−1)
6
-6
4
-4
Answer:
6
Step-by-step explanation:
Just easy lol have a good day
Which is greater -10 or -20 in absolute value
Determine the exact value of cos (11pi/6
8. Find the radius of a circular lid with an area of 12in.^2.
Divide. (18x^3+12x^2-3x)/6x^2
A)x^2+x-1/2x
B)3x+2
C)3x+2 -1/2x
D)3x -1/2x
Divide.(2x^3-x^2-24x+12)/(2x-1)
A)x^2-1
B)x^2-12
C)2x-1
D)x^2-1/2x-12
Can anybody help?
Answer:
1. The correct option is C.
2. The correct option is B.
Step-by-step explanation:
1.
The given expression is
[tex]\frac{18x^3+12x^2-3x}{6x^2}[/tex]
Simplify the expression,
[tex]\frac{18x^3}{6x^2}+\frac{12x^2}{6x^2}-\frac{3x}{6x^2}[/tex]
Cancel out the common factors.
[tex]3x+2-\frac{1}{2x}[/tex]
Therefore option C is correct.
2.
The given expression is
[tex]\frac{2x^3-x^2-24x+12}{2x-1}[/tex]
Use grouping method to simplify the numerator.
[tex]\frac{x^2(2x-1)-12(2x-1)}{2x-1}[/tex]
[tex]\frac{(2x-1)(x^2-12)}{2x-1}[/tex]
Cancel out the common factors.
[tex]x^2-12[/tex]
Therefore option B is correct.
you have been given the task of transporting 3000 apples ,1000 miles from apple land to bananaville. your truck can carry 1,000 apples at a time. every time you travel a mile towards bananaville you must pay a tax of 1 apple but you pay nothing when going in the other direction towards appleland.what is the highest number of apples you can get to bananaville?
For an investment of $26,245, a quarterly statement reports that the account balance is $26,292. The statement also reports that for the same quarter, the rate of return on the investment was -0.02%. Given the information regarding the investment's current balance, does the reported rate of return seem reasonable? Use complete sentences
Name the set(s) of numbers to which -2/5 belongs.
A. Whole numbers, integers, rational numbers
B. Rational numbers
C. Whole numbers, natural numbers, integers
D. Integers, rational numbers
josè and nancy take care of animals when the owners are away. josè charges $15 plus $2 per animal. nancy charges $5 plus $3 per animal. for how many animals do they charge the same amount? write a system of equation to model the problem and solve the system to find out.
José and Nancy would charge the same amount for taking care of 10 animals. This is found by solving the system of equations which represents their respective charges.
Explanation:The subject of this problem is determining for how many animals José and Nancy would charge the same amount. This is a classic algebra problem that can be solved by setting up a system of equations.
Let's designate 'x' as the number of animals. José's charges can be represented by the equation y = 15 + 2x, because he charges a flat fee of $15 and then an additional $2 per animal. Nancy's charges can be represented by the equation y = 5 + 3x, because her initial fee is $5, and she charges an additional $3 per animal.
To find out when they both charge the same amount, set these two equations equal to each other and solve for x:
15 + 2x = 5 + 3x
This simplifies to 10 = x, meaning they would both charge the same amount for 10 animals.
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Please help on this problem
Help me please I don’t understand
Help answer for a b and c work all of it out and show your work!thanks. Use the picture.
Answer:
Nah thx for the free pnts tho
Step-by-step explanation:
how do you find out the percent of the pie chart
Final answer:
To determine the percentage of a pie chart slice, divide the value by the whole and multiply by 100. The pie represents the entire data set, and the slices show the percentage that each category contributes to the whole.
Explanation:
To find out the percent of a pie chart, observe that each slice of the pie represents a specific share of the whole. If you know the entire value that the pie chart represents, you can calculate the percentage for each slice by dividing the value of the slice by the total value and multiplying by 100. For instance, if a pie chart represents the population by age groups, and there are 1000 people in total where 200 are aged 65 and over, the slice of the pie chart for those 65 and over would be (200/1000)*100, which equals to 20%.
Pie graphs are particularly useful for displaying how a whole is divided into parts, such as in division of government spending, categorization of the population, or distribution of market share among firms. A key point to remember is that the sum of all percentages in a pie chart should equal 100%, unless it is representing overlapping categories where participants can be counted in multiple categories. When that's the case, a pie chart is not the appropriate graph to use.
15 points!!!!!help!!!!!!!!!!!!
Which table correctly shows all the sample spaces for tossing a coin and rolling an odd number on a six-sided number cube? Coin Number cube Outcome Heads 1 Heads, 1 Heads 3 Heads, 3 Heads 5 Heads, 5 Tails 1 Tails, 1 Tails 3 Tails, 3 Tails 5 Tails, 5 Coin Number cube Outcome Heads 1 Heads, 1 Heads 2 Heads, 2 Heads 3 Heads, 3 Tails 4 Tails, 4 Tails 5 Tails, 5 Tails 6 Tails, 6 Coin Number cube Outcome Heads 2 Heads, 2 Heads 4 Heads, 4 Heads 6 Heads, 6 Tails 2 Tails, 2 Tails 4 Tails, 4 Tails 6 Tails, 6 Coin Number cube Outcome Heads 2 Heads, 1 Heads 4 Heads, 3 Heads 6 Heads, 5 Tails 2 Tails, 2 Tails 4 Tails, 4 Tails 6 Tails, 6
Answer:
Coin Number
cube Outcome
Heads
2
Heads, 2
Heads
4
Heads, 4
Heads
6
Heads, 6
Tails
2
Tails, 2
Tails
4
Tails, 4
Tails
6
Tails, 6
Step-by-step explanation:
Please HELPPPPPP !!
Triangle DEF is translated two units up and two units right on a coordinate plane. Which of the following scenarios best represent(s) triangles DEF and D’E’F’. Select all that apply.
The area of triangle D’E’F’ is twice the area of triangle DEF.
The area of triangle D’E’F’ is equal to the area of triangle DEF.
The area of triangle D’E’F’ is half the area of triangle DEF.
The area of triangle D’E’F’ is four times the area of triangle DEF.
Answer:
that is right its The area of triangle D’E’F’ is equal to the area of triangle DEF.
Step-by-step explanation:
1440 tons of grain were brought to a silo over two days. On the second day, 80% of the amount of the grain delivered during the first day was brought in. How many tons of grain were delivered to the silo on the first day?
Answer:
800
Step-by-step explanation:
An isosceles triangle has a base of 8 feet and a height of 6 feet. What is the length of one of the two congruent sides? Work and a diagram would be appreciated using the Pythagorean Theorem