The sum of a number and its square is 2. Find the number(s).
Answers
.. x^2 +x = 2
.. x^2 +x +1/4 = 9/4 . . . . add 1/4 to complete the square
.. x +1/2 = ±3/2 . . . . . . . .square root
.. x = -2, 1
Numbers meeting that requirement are -2 and 1.
The number 'x' that satisfies the condition of the sum of a number and its square equaling 2 are 1 and -2. This is determined by setting up a quadratic equation and solving for 'x'.
Explanation:To solve this problem, let's start by defining the unknown number we are looking for as 'x'. According to the problem, the sum of a number (x) and its square (x²) equals 2. As a mathematical equation, this would be written as x + x² = 2.
Once this equation is set up, we can solve for x. However, the equation is quadratic, which means it may have two solutions. Anyway, the equation can be rearranged into the standard quadratic format as x² + x - 2 = 0. It can be factored into (x - 1)(x + 2) = 0. This gives us two potential solutions for x: x = 1 and x = -2.
To verify if these solutions are valid, we can substitute them back into the original equation: 1 + 1² = 2 is true, and -2 + (-2)² = 2 is also true. Thus, both of these numbers, 1 and -2, are solutions to the original problem.
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Related Questions
Help please!!!! I dont know this.
Answers
The area of the semicircle is
.. (1/2)π(20 in)^2 = 200π in^2
The area of the triangle is
.. (1/2)*(24 in)*(32 in) = 384 in^2
The shaded area is the difference
.. (200π -384) in^2
There are 20 goldfish in a pond. Their population is increasing by 20% each year. The same pond has 100 minnows. The minnow population is increasing by 10 minnows each year. Make a graph to find the year that the two species of fish will have the same population. In what year will the fish populations be approximately the same?
Answers
We only consider the portion of the graph from x=0 on, since negative time is illogical. Tracing the graph we get the intersection point.
Answer:
13.5 years.
Step-by-step explanation:
We know that exponential growth function is in form [tex]y=a\cdot(1+r)^x[/tex], where, a is initial value and r is growth rate in decimal form.
Population of goldfish after x years would be [tex]y=20\cdot(1+0.20)^x[/tex].
We know that a linear function is in form [tex]y=mx+b[/tex], where, b is initial value and m is slope.
Population of minnow after x years would be [tex]y=100+10x[/tex].
Graphing both equations, we will get our required graph as shown in the attachment.
Since both graphs intersect at [tex]x=13.5[/tex], therefore, in the 13.5 years both populations will approximate the same.
A cone has a volume of 12 cubic inches. What is the volume of a cylinder that the cone fits exactly inside of?
6 in3
24 in3
36 in3
48 in3
Answers
Volume of a cylinder = r² π h = 3 · V ( cone ) = 3 · 12 = 36 in³
Answer: C
Solve the division problem. Round answer to the nearest hundredth. 9.252.063
Answers
Suppose f(x) = 0.125x for 0 < x < 4. determine the mean and variance of x. round your answers to 3 decimal places.
Answers
- mean: 2.667
- variance: 0.889
Explanation:
To get the mean and variance of x, we need to verify first whether...
- x is discrete or continuous random variable
- f is probability mass or probability density function
because if we cannot verify the 2 statements above, we can't compute the mean and the variance.
Since 0 < x < 4, x is a continuous random variable because x can be any positive number less than, which includes a non-integer.
Note that if the random variable is continuous and [tex]0 \leq f(x) \leq 1[/tex] for any values of x in the domain of f, then f is a probability density function (PDF).
Note that
[tex]0 \ \textless \ x \ \textless \ 4 \\ \Leftrightarrow 0.125(0) \ \textless \ 0.125x \ \textless \ 0.125(4) \\ \Leftrightarrow 0 \ \textless \ 0.125x \ \textless \ 0.5 \\ \Leftrightarrow 0 \ \textless \ f(x) \ \textless \ 0.5 \\ \Rightarrow 0 \ \textless \ f(x) \ \textless \ 1 (\text{Because }0 \ \textless \ f(x) \ \textless \ 0.5 \ \textless \ 1)[/tex]
Hence, for any x in the domain of f, 0 < f(x) < 1. Moreover, since x is a continuous random variable, thus f is a PDF.
First, we use the following notations for mean and variance:
E[x] = mean of x
Var[x] = variance of x
Since f is a probability density function, we can use the following formulas for the mean and the variance of x:
[tex]\boxed{\text{mean of }x = E[x] = \int_{-\infty}^{\infty}{xf(x)}dx}[/tex]
[tex]\boxed{\text{Variance of }x = \text{Var}[x] = E[x^2] - (E[x])^2}} [/tex]
To compute for the mean of x,
[tex]\text{mean of }x = E[x] \\ = \int_{-\infty}^{\infty}{xf(x)}dx} \\ = \int_{-\infty}^{\infty}{x(0.125x)}dx} \\ \boxed{\text{mean of }x = \int_{-\infty}^{\infty}{0.125x^2}dx}}[/tex]
The integral seems complicated because of the infinity sign. But because the domain of f is the set of positive numbers less than 4, that is,
[tex]\text{domain of }f = \left \{x : 0 \ \textless \ x \ \textless \ 4 \right \}[/tex]
the bounds of the integral for the mean can be changed from [tex]-\infty \ \textless \ x \ \textless \ \infty[/tex] to [tex]0 \ \textless \ x \ \textless \ 4[/tex] so that
[tex]\boxed{\text{mean of }x = \int_{-\infty}^{\infty}{0.125x^2}dx = \int_{0}^{4}{0.125x^2}dx}[/tex]
Hence, the mean is computed as
[tex]\text{mean of }x = \int_{0}^{4}{0.125x^2}dx \\ = \left[ \frac{0.125x^3}{3} \right]_{0}^{4} \\ \\ = \left[ \frac{0.125(4)^3}{3} \right] - \left[ \frac{0.125(0)^3}{3} \right] \\ \\ \boxed{\text{mean of }x = \frac{8}{3} \approx 2.667} [/tex]
Since the formula for variance is computed as
[tex]\text{Variance of }x = \text{Var}[x] = E[x^2] - (E[x])^2[/tex]
we must first compute for [tex]E[x^2][/tex] for which
[tex]E[x^2] = \int_{-\infty}^{\infty}{x^2 f(x)}dx \\ \\ = \int_{-\infty}^{\infty}{x^2(0.125x)}dx \\ \\ \boxed{E[x^2] = \int_{-\infty}^{\infty}{0.125x^3}dx}[/tex]
Similar to the computation of integral of the mean, we take note that
[tex]\text{domain of }f = \left \{x : 0 \ \textless \ x \ \textless \ 4 \right \}[/tex]
so that we can change the bounds of the integral, that is,
[tex]\boxed{E[x^2] = \int_{-\infty}^{\infty}{0.125x^3}dx = \int_{0}^{4}{0.125x^3}dx}[/tex]
Hence,
[tex]E[x^2] = \int_{0}^{4}{0.125x^3}dx \\ \\ = \int_{0}^{4}{0.125x^3}dx \\ \\= \left[ \frac{0.125x^4}{4} \right]_{0}^{4} \\ \\ = \left[ \frac{0.125(4)^4}{4} \right] - \left[ \frac{0.125(0)^4}{4} \right] \\ \\ \boxed{E[x^2] = 8}[/tex]
Because [tex]E[x] = \frac{8}{3} [/tex],
[tex]\text{Variance of }x \\ \\ = E[x^2] - (E[x])^2 \\ \\ = 8 - \left( \frac{8}{3} \right)^2 \\ \\ \boxed{\text{Variance of }x = \frac{8}{9} \approx 0.889}[/tex]
The mean of x is 1.35 and the variance of x is approximately 1.267.
Explanation:To find the mean of x, we need to calculate the expected value. We can do this by multiplying each value of x by its corresponding probability and then summing up the results. In this case, using the given values and probabilities, we have 0(0.20) + 1(0.45) + 2(0.20) + 3(0.10) + 4(0.05) = 1.35. Therefore, the mean of x is 1.35.
To find the variance of x, we need to calculate the squared difference between each value of x and the mean, weighted by their respective probabilities. We then sum up these values. Using the formula for variance, we have (0-1.35)²(0.20) + (1-1.35)²(0.45) + (2-1.35)²(0.20) + (3-1.35)²(0.10) + (4-1.35)²(0.05). By simplifying this expression, we find that the variance of x is approximately 1.267.
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The square footage of a house is 1200 square feet. What type of data is this?
Answers
The explanation is shown above:
1- By definition, the area is the measure of a two-dimensional surface.Its units can be written in m^2, ft^2, in^2...However, in the International System of Units is m^2.
2. This is very useful when you want to know the surface of a house. If it is a rectangular house, you can calculate the area as below:
A=LxW
Where L is the length and W is the width
1.3 to the second power
Answers
1.3^2 = 1.3 x 1.3 = 1.69
Hal wants to make a 2 ½ foot banner from a 5-foot length of cloth. If he has marked 2.0 on the cloth, what does he have to do to find 2 ½ feet?
Answers
---
Hope this helps!
Six girls share 5 pints of milk equally.What fraction of a pint of milk does each girl get?
Answers
All you have to do is divide 6 by 5.
Two friends mix blue paint and yellow paint to make batches of green paint, as shown in the tables. Jarrod Cups Blue Cups Yellow 3 2 Ian Cups Blue Cups Yellow 5 2 Which correctly compares their ratios of blue to yellow paint?
Answers
Cups Blue Cups Yellow
3 2
Ian:
Cups Blue Cups Yellow
5 2
We have that their ratios are:
Jarrod: 3/2
Ian: 5/2
Comparing both ratios:
Jarrod uses 60% yellow paint and 40% blue paint.
Ian uses 71.4% yellow paint and 28.4% blue paint.
Answer:
Jarrod: 3/2
Ian: 5/2
What is the answer for # 24?
A B C or D
Answers
The volume of the space not filled by the sphere is the difference between the volume of a cube with edge length 6 inches and the volume of a sphere with radius 3 inches.
CubeThe volume of a cube of edge length s is
... V = s³
When the edge length is 6 in, the volume is
... V = (6 in)³ = 216 in³
SphereThe volume of a sphere with radius r is
... V = (4/3)π·r³
When the radius is 3 in, the volume is
... V = (4/3)π·(3 in)³ = 36π in³
SpaceThen the volume of the space between the cube and the sphere is
... Vcube - Vsphere = 216 in³ - 36π in³ ≈ 102.9 in³ . . . . corresponding to choice C
Ben has 1/2 of a loaf of bread if he and his 3 friends share the 1/2 loaf equally how much of the whole loaf does each person get?
Answers
1/2 loaf of bread
3 friends
1/2 ÷ 3 ⇒ 1/2 * 1/3 = 1*1 / 2*3 = 1/6
In dividing fractions, get the reciprocal of the divisor and multiply it to the dividend
1/2 ÷ 3
or
1/2 ÷ 3/1
To divide fractions use Copy Dot Flip
Copy
1/2 ÷ 3/1
Dot
1/2 · 3/1
Flip
1/2 · 1/3
1/6 is your answer
how to solve this word problem
Answers
.. (number in school)/(560 lunches sold) = (60 students sampled)/(40 eat lunch)
Then
.. (number in school) = 560*60/40 = 840
For a science experiment,Juanita records the height of a plant every day in centimeters.What is the attribute measured in her experiment.
Answers
In Juanita's science experiment, the attribute measured is the plant's height, a key phenotypic trait in studying plant growth, which she records in centimeters, using consistent and precise units for scientific reliability.
The attribute measured in Juanita's science experiment is the height of a plant, which is a phenotypic trait important in plant biology. Juanita is collecting data related to plant growth by observing and recording the plant's height in centimeters daily. This meticulous tracking allows for analysis of plant growth patterns over time and the effect of various treatments, such as different fertilizers or amounts of water. The typical unit of measurement for the height of a plant is in centimeters or meters, which are part of the metric system used in scientific research.
During a laboratory experiment like the one Juanita is conducting, it is essential to measure such variables with precision and consistency. Measurements should be recorded using uniform units and methods to ensure reliability in the results, which may include the plant height, recorded in centimeters, at regular intervals. A graph representing this data might portray the number of days since the start of the experiment along the X-axis and the plant's height along the Y-axis, thereby allowing a visual assessment of the plant's growth over time.
A bag contains 2 red balls and 18 green balls. A ball is chosen at random from the bag. What is the BEST answer for the probability of drawing a green ball?
Answers
18 green
18/20 = 9/10 probability
A real estate agent knows that he will receive a commission of $4,250 from the sale of a property when the deal is completed 37 days from now. Needing cash today, though, he signs a discount note at a credit union, using his expected commission as the maturity value. The discount rate is 9.55%. Find the effective rate (APR).
Answers
.. effective rate = r/(1-rt)
Assuming a 365 day year, this is
.. = 9.55%/(1 -0.0955*(37/365))
.. ≈ 9.64%
You are deciding on two different designs for envelopes.
a. Which design has the greater area?
The area of the first design is
52 square inches and the area of the second design is
61.375 square inches.
b. You make 500 envelopes using the design with the greater area. Using the same amount of paper, how many more envelopes can you make with the other design?
Please answer part b with the information given.
-A student in need
Answers
The area of the two designs are given as 52 square inches and 61.375 square inches.
In these the second design 61.375 square inches is having the greater area.
If 500 envelops are made then area of paper = 61.375x500 =30687.5 square inches.
If envelops having 52 square inches are made out of 30687.5 square inches of paper ,number of envelops made =30687.5 ÷ 52=590.144
590 small envelops can be made having area as 52 square inches.
It is 590-500=90 more than the larger envelops made.
2(Square root 80/5 -5) = ??
Please help and show work! =^-^=
Answers
Final answer:
To solve the expression 2(√(80/5) - 5), we simplify the square root first, then subtract 5, and finally multiply by 2, which results in the solution -2.
Explanation:
To solve the expression 2(√(80/5) - 5), let's first simplify the square root and the division inside it.
We have √(80/5) = √16 = 4, since 80 divided by 5 is 16, and the square root of 16 is 4.
Next, we subtract 5 from 4: 4 - 5 = -1.
Finally, we multiply the result by 2: 2 * (-1) = -2.
Therefore, the solution to the expression 2(√(80/5) - 5) is -2.
The answer to the given question is: [tex]\[ 2\left(\frac{\sqrt{80}}{5} - 5\right) = \frac{2\sqrt{80}}{5} - 10 \][/tex]
To solve the expression step by step:
1. First, simplify the square root within the parentheses. The number 80 can be factored into [tex]\( 16 \times 5 \)[/tex], and since [tex]\( \sqrt{16} = 4 \),[/tex] we can rewrite[tex]\( \sqrt{80} \) as \( 4\sqrt{5} \).[/tex]
[tex]\[ 2\left(\frac{4\sqrt{5}}{5} - 5\right) \][/tex]
2. Next, distribute the 2 across the terms inside the parentheses:
[tex]\[ 2 \times \frac{4\sqrt{5}}{5} - 2 \times 5 \][/tex]
3. Simplify each term:
[tex]\[ \frac{8\sqrt{5}}{5} - 10 \][/tex]
4. At this point, we have the simplified form of the expression. However, if we want to combine the terms into a single fraction, we need a common denominator. The second term, -10, can be written as [tex]\( \frac{-10 \times 5}{5}[/tex]) to get the common denominator of 5:
[tex]\[ \frac{8\sqrt{5}}{5} - \frac{50}{5} \][/tex]
5. Now, subtract the two fractions:
[tex]\[ \frac{8\sqrt{5} - 50}{5} \][/tex]
6. This is the final simplified form of the expression. It cannot be simplified further algebraically because [tex]\( 8\sqrt{5} \)[/tex] and 50 do not have common factors other than 1.
Therefore, the final answer is: [tex]\[ \boxed{\frac{8\sqrt{5} - 50}{5}} \][/tex]
[tex]\[ 2\left(\frac{\sqrt{80}}{5} - 5\right) \][/tex]
And the final simplified answer is: [tex]\[ 2\left(\frac{\sqrt{80}}{5} - 5\right) = \frac{2\sqrt{80}}{5} - 10 \][/tex]
HELP PLEASE , THANK YOU IF YOU DO!!!!
Answers
5 goes right 5 units and that's where you begin.
-6 goes left and that's where you end up.
The answer is minus 1.
A number line works best when you are dealing with 1 sign between numbers. Two signs will only confuse you.
At noon, ship a is 50 km west of ship
b. ship a is sailing south at 10 km/h and ship b is sailing north at 20 km/h. how fast is the distance between the ships changing at 4:00 pm? (round your answer to one decimal place.) 23.05 incorrect: your answer is incorrect. km/h
Answers
At 4:00 pm, the distance between the ships is changing at a rate of 30 km/h.
Explanation:To find how fast the distance between the two ships is changing, you can use the Pythagorean theorem, as the two ships are moving at right angles to each other (north and south). Let's denote the distance between the two ships as "D," the speed of ship A as "vA," and the speed of ship B as "vB."
At any given time, you have:
D² = (Distance ship A travels)² + (Distance ship B travels)²
Now, we can differentiate both sides of this equation with respect to time "t" to find how the distance "D" is changing:
2D * dD/dt = 2(vA * dA/dt) + 2(vB * dB/dt)
Here, dD/dt represents the rate of change of the distance between the ships, dA/dt is the speed of ship A (10 km/h), and dB/dt is the speed of ship B (20 km/h).
Plug in the values:
2 * (dD/dt) = 2 * (10 km/h) + 2 * (20 km/h)
Now, solve for dD/dt:
2 * (dD/dt) = 20 km/h + 40 km/h
2 * (dD/dt) = 60 km/h
Now, divide by 2:
dD/dt = 60 km/h / 2 = 30 km/h
So, at 4:00 pm, the distance between the two ships is changing at a rate of 30 km/h.
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A person earns 26,800 one year and gets a 5% raise in salary. What is the new salary?
Answers
The new yearly salary is $28,140.
Explanation:Amount of raise.
5%=0.05
Multiply the original salary times the percentage of the raise.
$26,800×0.05=$1,340
New salary .
$26,800+1,340=$28,140
HOPE THIS HELPS! :)
This question is based on the concept of percentage. Therefore new salary when there is 5% raise in 26,800 is 28,140. There are many uses of calculating percent.
What is percent ?Percent is a proportionate of a number. Percent is a dimensionless quantity that is it has no unit. Percent is always out of hundred whereas fraction is always from 1. It is represented with the sign %. There are many uses of calculating percent. It is used in calculating interest in bank. It used in calculating inflation rate.
Mathematically,
The new yearly salary of a person is 28,140.
percentage of raise in salary is
5%=0.05
Multiply the original salary and the percentage of the raise in salary
26,800×0.05=1,340
New salary =26,800+1,340=28,140
Therefore new salary when there is 5% raise in 26,800 is 28,140
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Prime numbers have less 1.than two factors.
2.more than two factors. 3.exactly two factors. 4.less than or equal to two factors.
Answers
exactly 2 factors
Answer:
They just have 2 factors :/
Step-by-step explanation:
Bc yes
what is -6x = -24
13 points
Answers
.. (-6x)/-6 = (-24)/-6
.. x = 4
_____
You asked "what is ..."
Answer: it is a "one-step" linear equation in x.
a customer deposits $2000 in a savings account that pays 5.2% interest compounded continuously. how much will be earned between the 3rd and 5th yeards?
Answers
FV=p(1+r/100)^n
where:
FV=future value
r=rate
n=time
The amount earned in the first 3 years will be:
FV=2000(1+5.2/100)^3=$2,328.50
The amount earned in 5 years will be:
FV=2000(1+5.2/100)^5=$2,576.97
The amount earned between 3rd and 5th
=2,576.96-2,328.50
=$248.46
G let x be an exponentially distributed random variable with parameter λ = 1 / 2 . determine the probability distribution function of the random variable y = x 2 . what kind of distribution does y have?
Answers
[tex]F_X(x)=\mathbb P(X\le x)=\begin{cases}1-e^{-\lambda x/2}&\text{for }x\ge0\\0&\text{otherwise}\end{cases}[/tex]
The CDF of [tex]Y[/tex] is then
[tex]F_Y(y)=\mathbb P(Y\le y)=\mathbb P(X^2\le y)=\mathbb P(X\le\sqrt y)=F_X(\sqrt y)[/tex]
[tex]\implies F_Y(y)=\begin{cases}1-e^{-\lambda\sqrt y/2}&\text{for }y\ge0\\0&\text{otherwise}\end{cases}[/tex]
[tex]\implies F_Y(y)=\begin{cases}1-e^{-(y/(4/\lambda^2))^{1/2}}&\text{for }y\ge0\\0&\text{otherwise}\end{cases}[/tex]
which is the CDF of a Weibull distribution with shape parameter [tex]\dfrac4{\lambda^2}[/tex] and scale parameter [tex]\dfrac12[/tex].
Find the area of the triangle that divides the parallelogram in half
Answers
If ab = 8 and a^2+b^2=16, then what is the value of (a+b)^2
Answers
= 16 + 2 (8)
= 16 + 16
= 32
Attached is my work.
A high school has 3636 players on the football team. the summary of the players' weights is given in the box plot. approximately, what is the percentage of players weighing greater than or equal to 194194 pounds?
Answers
To calculate the percentage of players weighing 194 pounds or more from a box plot, we need to know where 194 pounds falls in relation to the quartiles on the plot. Without the box plot details, we cannot perform the needed analysis.
The student's question appears to concern the analysis of a box plot depicted in their materials to find the percentage of players weighing 194 pounds or more on a football team. However, to accurately answer this question, we need the specific details from the box plot, including the median (Q2), first quartile (Q1), third quartile (Q3), and any outliers if provided. From the box plot, we would determine where the weight of 194 pounds falls in relation to the quartiles. If 194 pounds is at or above Q3, then you would expect a lower percentage of players above this weight; if it is below Q3, then the percentage could be higher.
The approach typically involves identifying the number of players within each quartile and calculating the corresponding percentages. Without the specific box plot data, we cannot perform these calculations. Should you provide the box plot details, we can review the quartiles and calculate the exact percentage of players weighing 194 pounds or more.
The polynomial 8x2 – 8x + 2 – 5 + x is simplified to 8x2 – gx – h. What are the value of g and h?
g = –9 and h = 7
g = 9 and h = –3
g = –7 and h = 7
g = 7 and h = 3
Answers
Answer:
Its D i just took the test
Step-by-step explanation:
Answer:
D. g=7 and h=3
Step-by-step explanation:
What are the zeros of the polynomial function?
f(x)=x^2+9x+20
Enter your answers in the boxes.
Answers
9^2-4*1*20=1.
Using the quadratic formula, we get the solutions:
[tex] \frac{-9-1}{2} =-5\\ \frac{-9+1}{2}=-4 [/tex]
the solutions are then -4 and -5.