Mathematics College

find the discriminant & describe the nature of the roots 2x2=3x+5

Answers

Answer 1
[tex]\bf 2x^2=3x+5\implies 2x^2-3x-5=0 \\\\\\ \begin{array}{llccll} &{{ 2}}x^2&{{ -3}}x&{{ -5}}&=0\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array} \\\\\\ discriminant\implies b^2-4ac= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases}\\\\ -------------------------------\\\\ (-3)^2-4(2)(-5)[/tex]

Related Questions

A family pass to amusement park cost $54 using the distributive property write an expression that can be used to find the cost in dollars of 8 family

Answers

8(50+4)

I hope this helped! :))

Find the intervals of convergence for the function. f(x)=sum_(n=1)^infinity x**n/n**2 32324243232 incorrect: your answer is incorrect. (b) find the intervals of convergence for f '.

Answers

[tex]\displaystyle\sum_{n\ge1}\frac{x^n}{n^2}[/tex]

will converge as long as

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|<1[/tex]

according to the ratio test. We have

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{(n+1)^2}}{\frac{x^n}{n^2}}\right|=|x|\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=|x|\left(\lim_{n\to\infty}\frac n{n+1}\right)^2=|x|<1[/tex]

so the sum will certainly converge whenever [tex]-1<x<1[/tex], but we also know the series will converge at the endpoints. [tex]\displaystyle\sum\frac1{n^2}[/tex] is a convergent [tex]p[/tex]-series, which means [tex]\displaystyle\sum\frac{(-1)^n}{n^2}[/tex] is also (absolutely) convergent, so in fact the interval of convergence is [tex]-1\le x\le1[/tex] (endpoints included).

A rectangular field is to be bounded by a fence on 3 sides and a straight stream on the 4th side. Find the dimensions of the field with maximum area that can be enclosed using 1000 ft of rope. This is in the applied maximum and minimum chapter btw.

Answers

check the picture below.

[tex]\bf \stackrel{area}{A}=length\cdot width\implies A(w)=(1000-2w)w \\\\\\ \boxed{A(w)=1000w-2w^2} \\\\\\ \cfrac{dA}{dw}=1000-4w\implies 0=1000-4w\implies 4w=1000 \\\\\\ w=\cfrac{1000}{4}\implies \boxed{w=\stackrel{width}{250}} \\\\\\\ [1000-2(250)]\implies 1000-500\implies \boxed{l=\stackrel{length}{500}}[/tex]

and if you run a first-derivative test on say x = 249 and x = 251, you'll notice the slope is positive and negative respectively, meaning, is a maximum.
Final answer:

To maximize the area of the rectangular field, express the area in terms of one variable, derive the area function, and set it to zero for solving. The sides of the field yielding the maximum area are obtained using these calculations.

Explanation:

This question involves optimization in calculus, specifically related to finding the maximum area of a rectangle. In such a problem, the first step is to express the area (A) of the rectangle in terms of one variable. Given a fence that is 1000 ft in length, if we name the side perpendicular to the stream as 'x', and the side parallel to it (bounded by the fence) as 'y', the perimeter of the field is 2x + y = 1000.

We can express y in terms of x: y = 1000 - 2x. The area of the rectangle, A = x*y = x*(1000 - 2x).

To find the maximum area, we find the first derivative of A with respect to x, set it to zero and solve for x. The second derivative test will confirm if it's a maximum. Based on the solutions, the maximum area is achieved with the dimensions of the rectangular field being specific values of x and y that you compute.

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write 129.04 in words

Answers

One Hundred Twenty Nine and Four Hundredths
One hundred twenty nine and four hundreds

Smith High School offers a baseball camp that is $75 for 4 days of camp and a basketball camp that is $100 for 5 days of camp. Which camp is a better deal and by how much?

Answers

For the baseball camp, you would be paying $18.75 per day. For the basketball camp, you would be paying $20 per day. Judging by the prices, you would be more likely to have a better price for the baseball camp by 1.25. Yes, it isn't that much, but it's still worth it. Hope it helped!

Merritt is driving to mount Shasta. On her map, she is a distance of 7.75 inches away. The scale on the map is .5 inch = 50 Miles. How far must Merritt travel to reach her destination?

Answers

193.75 miles cause 7.75 * .5 = 3.875 and 3.875* 50 = 193.75

(−5k
​3
​​ −6k+1)+(−6k
​2
​​ +5k)

Answers

Remove unnecessary parenthesis.

-5k 3 - 6k +1 - 6k 2 + 5k (simplify each term) -15k - 6k + 1 - 12k +5k = -28k +1

Answer: 28k + 1

Step-by-step explanation:

−5k

3

−6k

2

−k+1

If x has a continuous uniform distribution on the interval (0,5), what is the 25th percentile for x?

Answers

Given that [tex]X\sim\mathrm{Unif}(0,5)[/tex], it has PDF

[tex]f_X(x)=\begin{cases}\dfrac15&\text{for }0\le x\l5\\\\0&\text{otherwise}\end{cases}[/tex]

and thus CDF

[tex]F_X(x)=\mathbb P(X\le x)=\begin{cases}0&\text{for }x<0\\\\\dfrac x5&\text{for }0\le x<5\\\\1&\text{for }x\ge5\end{cases}[/tex]

The 25th percentile is the value [tex]X=k[/tex] such that [tex]\mathbb P(X\le k)=0.25[/tex]. We have

[tex]\mathbb P(X\le k)=\dfrac k5=0.25\implies k=1.25[/tex]

Suppose the concentration of a certain pollutant is uniformly distributed in the range of 4 to 20 ppm (parts per million). if a concentration in excess of 15 ppm is considered toxic, what is the probability that a specimen will yield a toxic concentration level

Answers

Uniform probability?  Here the probability function graph is a rectangle of length 20-4 (or 16) and width / height of 1/16.  Note that the area under the graph from x=4 to x=20 is 1 (which is always the case when we're discussing probability functions).  The area under the graph from 15 to 20 is (20-15)(1/16), or
5(1/16), or 5/16.  Thus, the probability that a specimen will yield a toxic concentration level is 5/16.

I have no way of knowing what you already know about probability distributions.  If you need a more detailed explanation, please ask questions here.

What is the distributive property of 27 + 60

Answers

You could distribute them by 60 ( 27+30+27+30) or by 27 (60+20+7)


Hoped it helped! :)

Kesha sharpened 4 colors of pencils for her teacher. She sharpened equal amounts of red and green pencils. Then she sharpened twice as many blue and yellow pencils in equal amounts. How many of the 42 pencils were yellow?

Answers

Variables are taken from the first letter of the color, ex: green = g.

Given:
r = g
2r = b = y
Total =42

So then:
r+ r + 2r + 2r = 42
6r = 42
r = 7

y = 2r
y = 2(7)
14 of the 42 pencils were yellow.

Answer:

Step-by-sstep explanation:

dr. lee went to a bank to get a loan for some new furniture for the waiting area at her clinic. she wanted to borrow $5,000 to be paid back over 3 years. last week the interest rate was 4.25%. but this week theinterest rate went up to 4.50%. how much additional interest will Dr. lee pay because she waited the extra week

Answers

5000*(4.50%-4.25%)= 5000*0.25%=12.5 dollars

additional interest is 12.5 dollars.

A jar holds 128 fluid ounces of juice. The label says the jar has 16 servings. How many fluid ounces are needed for 80 servings?

Answers

There are 16 servings for each 128 fluid ounces of juice. 

Divide 80 servings by 16 servings, and 5 should be the quotient. Therefore, there are 5 jars of juice. Multiply 128 by 5 since 128 fluid ounces are in one jar. The answer should be 640 fluid ounces. 

Answer: 640 fluid ounces

Step-by-step explanation:

We assume that the fluid ounces in jar is directly proportional to the number of servings.

Given : A jar holds 128 fluid ounces of juice. The label says the jar has 16 servings.

Ratio of  fluid ounces of juice to number of servings =[tex]\dfrac{128}{16}[/tex]  (1)

Let x denotes the fluid ounces are needed for 80 servings.

Ratio of  fluid ounces of juice to number of servings =[tex]\dfrac{x}{80}[/tex]   (2)

From (1) and (2) , we have

[tex]\dfrac{x}{80}=\dfrac{128}{16}[/tex]   [in direct variation ration  remains constant.]

[tex]x=\dfrac{128\times80}{16}=640[/tex]

Hence, 640 fluid ounces are needed for 80 servings.

Find all the real fourth roots of 256.


4
-8 and 8
8
-4 and 4

Answers

256 = 16^2.  In turn, 16=4^2.

The 4th roots are {4, 4, -4 and -4}.

Check for yourself:  Does 4^4= 256?  does (-4)^4 = 256?

Rachel bought a book for $5.49 And a game for $10.98. She paid with a $20 bill. Which is the best estimate of the amount of change she should receive?

Answers

Rachel will have about $3.53 in change---------
So,
We can round 10.98 to 11 
We can round 5.49 to 5.50
11+5.50 = 16.50
20 - 16.50 = 3.5
So, the estimate of the amount of change Rachel would receive is $3.50.

Match the pairs of equations that represent concentric circles. Tiles

Answers

I will explain you and pair two of the equations as an example to you. Then, you must pair the others.

1) Two circles are concentric if they have the same center and different radii.

2) The equation of a circle with center xc, yc, and radius r is:

(x - xc)^2 + (y - yc)^2 = r^2.

So, if you have that equation you can inmediately tell the coordinates of the center and the radius of the circle.

3) You can transform the equations given in your picture to the form (x -xc)^2 + (y -yc)^2 = r2 by completing squares.

Example:

Equation: 3x^2 + 3y^2 + 12x - 6y - 21 = 0

rearrange: 3x^2 + 12x + 3y^2 - 6y = 21

extract common factor 3: 3 (x^2 + 4x) + 3(y^2 -2y) = 3*7

=> (x^2 + 4x) + (y^2 - 2y) = 7

complete squares: (x + 2)^2 - 4 + (y - 1)^2 - 1 = 7

=> (x + 2)^2 + (y - 1)^2 = 12 => center = (-2,1), r = √12.

equation: 4x^2 + 4y^2 + 16x - 8y - 308 = 0

rearrange: 4x^2 + 16x + 4y^2 - 8y = 308

common factor 4: 4 (x^2 + 4x) + 4(y^2 -8y) = 4*77

=> (x^2 + 4x) + (y^2 - 2y) = 77

complete squares: (x + 2)^2  - 4 + (y - 1)^2 - 1 = 77

=> (x + 2)^2 + (y - 1)^2 = 82 => center = (-2,1), r = √82

Therefore, you conclude that these two circumferences have the same center and differet r, so they are concentric.

Find the derivative of the function F(x) = (x4 + 9x2 − 5)4

Answers

The answer is [tex] 16x^{3} [/tex]

The required derivative of the function (x⁴ + 9x² − 5)⁴ is f'(x) = 4 (x⁴ + 9x² − 5)³(4x³ + 18x).

As per the question we have to determine the derivative of the f(x) = (x⁴ + 9x² − 5)⁴.

What is derivative?

A derivative is defined as the rate of change of the function with respect to the respective variables.

Here,
According to the question,
f(x) = (x⁴ + 9x² − 5)⁴
Differentiate both sides with respect to x.
f'(x) = d[(x⁴ + 9x² − 5)⁴] / dx
f'(x) = 4 (x⁴ + 9x² − 5)³ × (4x³ + 9×2x)
f'(x) = 4 (x⁴ + 9x² − 5)³(4x³ + 18x)


Thus, the required derivative of the function (x⁴ + 9x² − 5)⁴ is f'(x) = 4 (x⁴ + 9x² − 5)³(4x³ + 18x).

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Write the value of 1710 in standard form

Answers

Are y ou sure there's not more to this problem than you have shared?

1710 is already in simplest, standard form.

Sometimes you'll see it written as 1,710, which for some people is easier to read.

Colin drove 45 minutes to the airport. He arrived 90 minutes before his flight departed, and then he spent 70 minutes in the air. Once he landed, Colin spent 20 minutes gathering his luggage, and then he drove 35 minutes to his hotel. What must be true of any expression that represents the total time that Colin spent traveling from his house to the hotel? The numbers can be added in any order. The number 45 must be the first addend, and 35 must be the last addend. The numbers must be added in the order 45, 90, 70, 20, and 35. The numbers must be added in either the order 45, 90, 70, 20, and 35 or the order 35, 20, 70, 90, and 45.

Answers

Answer:

The numbers must be added in the order 45, 90, 70, 20, and 35.

Step-by-step explanation:

Answer:

The numbers must be added in the order 45, 90, 70, 20, and 35

Step-by-step explanation:

:3

During a catered lunch, an average of 4 cups of tea are poured per minute. The lunch will last 2 hours. How many gallons of tea should the caterer bring if there are 16 cups in one gallon?

Answers

4 cups every minute means 240 cups every hour:
4 X 60 (minutes per hour)

The lunch last 2 hours so if 240 cups is every hour then 480 cups would be for 2 hours.

Now covert 480 cups to gallons.  It takes 16 cups to make 1 gallon.  We would divide 480/16 = 30 gallons

The volume of a rectangular prism can be found by multiplying the base area, B, times the height. If the volume of the prism is represented by 15x2 + x + 2 and the height is x2, which expression represents B, the area of the base?

Answers

If you really meant that the height is x^2, then the area of the base is 

15x^2 + x + 2
------------------ .
         x^2

If you perform this division, you'll obtain 15 + (x+2)/(x^2).

So, again, verify your representation of the height of the prism.

It'd be good to have the multiple choice solutions to look at because it could be simplified.
          
Bx^2  = 15x^2 + x + 2
B = (15x^2 + x + 2) / x^2 
or simplified like this
B = 15x^2/x^2 + x/x^2 + 2/x^2
B = 15 + 1/x + 2/x^2

Find the extreme values of the function f(x, y) = 4x + 8y subject to the constraint g(x, y) = x2 + y2 − 5 = 0.

Answers

[tex]L(x,y,\lambda)=4x+8y+\lambda(x^2+y^2-5)[/tex]

[tex]L_x=4+2\lambda x=0[/tex]
[tex]L_y=8+2\lambda y=0[/tex]
[tex]L_\lambda=x^2+y^2-5=0[/tex]

[tex]L_y-2L_x=2\lambda y-4\lambda x=2\lambda(y-2x)=0\implies y=2x[/tex]

[tex]xL_x+yL_y=2\lambda(x^2+y^2)+4x+8y=10\lambda+4x+8y=0\iff5\lambda+2x+4y=0[/tex]

[tex]y=2x\implies 5\lambda+5y=0\implies 5(\lambda+y)=0\implies y=-\lambda[/tex]

[tex]4+2\lambda x=0\iff4+\lambda y=0\iff4-y^2=0\implies y=\pm2[/tex]
[tex]y=\pm2\implies x=\dfrac y2=\pm1[/tex]

In other words, there are two critical points, [tex](1,2)[/tex] and [tex](-1,-2)[/tex]. At the first point, there is a maximum of [tex]f(1,2)=20[/tex] and at the second, there is a minimum of [tex]f(-1,-2)=-20[/tex].

A rumour spreads exponentially through a college. 100 people have heard it by noon, and 200 by 1 p.m. How many people have heard it (a) by 3 p.m., (b) by 12.30 p.m., (c) by 1.45 p.rn.?

Answers

a) people have heard it  by 3 p.m is 800 people

b) people have heard it  by 12.30 p.m is 141 people

c) people have heard it  by 1.45 p.m is 530 people

Let's analyze this rumor spread problem. Each person who hears the rumor is assumed to spread it to exactly two other people.

a) By 3 p.m.: The rumor spreads for 3 hours after noon. Starting with 100 people at noon, by 3 p.m. it would be 100 * 2^3 = 100 * 8 = 800 people.

b) By 12:30 p.m.: The rumor spreads for half an hour after noon. That's 30 minutes or 0.5 hours. Since the rumor spreads exponentially, in half an hour, the number of people who hear it doubles once. So, by 12:30 p.m., it would be 100 * 2^0.5 = 100 * √2 ≈ 100 * 1.41 ≈ 141 people.

c) By 1:45 p.m.: The rumor spreads for 1 hour and 45 minutes after noon, which is 1.75 hours. So, by 1:45 p.m., it would be 100 * 2^1.75 ≈ 100 * 5.3 ≈ 530 people.

In summary:

a) 800 people

b) 141 people

c) 530 people

Andreus made $625 mowing yards. He worked for 5 consecutive days and earned the same amount of money each day. How much money did Andreus earn per day

Answers

625 / 5 = 125 per day

ally and barbara, and katherine will share the cost of a vacation rental for a week. aly agrees to pay 30% of the cost. barbara agrees to pay 0.45 of the coat. katherine will pay the remaining balance. if the total rental coast $960, how much will ally, barbara, and katherine each pay towards the week's rent?

Answers

Hi there! The total cost of the vacation is $960. Ally will pay 30% of the vacation rent. 960 * 0.3 is 288. Barbara will pay 45% of the cost. 960 * 0.45 is 432. Katherine will pay the remaining 25%. 960 * 0.25 is 240. Here are the prices that each of the women will pay.

Ally: $288
Barbara: $432
Katherine: $240

Answer with sometimes, always, or never true. Give an example/counterexample.

When a point is reflected across the y-axis, the new point has a negative x-coordinate.

Answers

Sometimes
If the original x coordinate is negative, it will become positive. For example, (-5, 5) would become (5,5), which is a positive x. The sign of the x changes when reflected over the y axis.

Final answer:

The statement that a point reflected across the y-axis results in a point with a negative x-coordinate is sometimes true, depending on the original point's location relative to the y-axis.

Explanation:

When a point is reflected across the y-axis, the result depends on the initial position of the point relative to the y-axis. The statement that the new point has a negative x-coordinate is sometimes true. If a point lies to the right of the y-axis (positive x-coordinate), reflecting it across the y-axis will indeed result in a point with a negative x-coordinate. However, if the original point is to the left of the y-axis (already has a negative x-coordinate), reflecting it across the y-axis will give it a positive x-coordinate.

Example

If the original point is (3, 2), reflecting it across the y-axis results in (-3, 2), which supports the statement.

If the original point is (-4, 5), reflecting it across the y-axis yields (4, 5), which contradicts the statement.

Therefore, the correct answer is sometimes true, demonstrating the importance of considering the position of the original point in relation to the y-axis.

Which of the following is the best example of a characteristic of an experiment?

a.Participants should be selected into the treatment group randomly.
b.Participants are rarely aware that they are part of an experiment.
c.Participants are observed without researchers making changes to their behaviors.
d.The results of an experiment are based on a questionnaire given to a random sample of the population.

Answers

The correct answer is C 
The correct option is A.
One of the basic feature of an experiment is the selection of participants by means of random method. It is very important to select participants randomly in order to avoid experimental bias that may occur as a result of the participants not been an exact representation of the concerned population.

What is the sales tax on a jacket priced at $350 if the sales tax rate is 7%

Answers

350 x .07 = 24.50

Your sale tax for a jacket would be $24.50.

Which linear function has the same y-intercept as the one that is represented by the graph?

Answers

The answer would be C because the y-intercept is when x is equal to 0
y-intercept when x = 0
on the graph x = 0 , y = 3

function
y = 2x + 3 when x = 0
y = 3

answer
C. y = 2x + 3

Is 9/6, 21/14 proportional?

Answers

Final answer:

The fractions 9/6 and 21/14 are proportional because both simplify to the same value, which is 1.5. Cross-multiplication also confirms their proportionality as the cross products are equal.

Explanation:

To determine if the fractions 9/6 and 21/14 are proportional, we can simplify each fraction or cross-multiply to see if we end up with an equivalent value. If two ratios are proportional, this means they are equivalent to each other when reduced to simplest form, or the cross products are equal.

Simplifying 9/6, we get 1.5, and simplifying 21/14, we also get 1.5. Since both simplified fractions result in the same value, 9/6 and 21/14 are indeed proportional.

We can also cross multiply to verify: (9×14) should be equal to (6×21). This results in 126 = 126, confirming that both ratios are indeed proportional.

Final answer:

The ratios 9/6 and 21/14 are proportional because both simplify to the same ratio 3/2, and cross-multiplication yields equal products, confirming proportionality.

Explanation:

To determine if the ratios 9/6 and 21/14 are proportional, we simplify each ratio. The ratio 9/6 simplifies to 3/2 and the ratio 21/14 simplifies to 3/2 as well. Since both simplified ratios are equal, we can confirm that 9/6 and 21/14 are indeed proportional ratios.

Another way to check for proportionality is to cross-multiply and see if the products are equal. Multiplying 9 times 14 and 6 times 21 gives us 126 in both cases, which reaffirms our conclusion. Thus, based on these methods, the given ratios are proportional.

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