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Here's another way

x^2 - 10 = 15       subtract 15 from both sides

x^2 - 25 =  0         factor  as a difference of squares

(x - 5)  (x + 5) =  0        set each factor to 0 and solve for x and we get that

x = 5     or    x  =  -5

I'm sorry your answer is incorrect.

A constant function, which is perfectly legit, maps the same value to every point in it's domain.

What is not allowed is mapping multiple values to a single point in the domain.

What we have described in the problem is a constant function with constant value of 5.

+10 on both sides

so it would be x^2=25

sqrt(25)= x

x= 5 

For completeness I'll show you how this is solved in general.  It's not trivial

\(\text{first we construct the matrix}\\ X = \begin{pmatrix}1 &x_1 &x_1^2\\1 &x_2 &x_2^2\\\vdots\\1 &x_n &x_n^2\end{pmatrix}\)

\(\text{In this problem this is}\\ X=\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \\ 1 & 4 & 16 \\ 1 & 5 & 25 \\ 1 & 6 & 36 \\ \end{array} \right)\)

\(\text{We then form the matrix}\\ Y=\begin{pmatrix}y_1\\y_2\\\vdots \\ y_n\end{pmatrix}\\ \text{In this problem this is}\\ (32, 78, 178, 326, 390, 337)^T\)

\(\text{The equation for the coefficients of the least squares quadratic fit is given by}\\ c = (X^TX)^{-1}X^TY\)

\(\text{Grinding this through we get}\\ c =(-143.9,154.418,-11.4107)^T \text{ i.e. that the best fit quadratic is}\\ y(t) = -11.41t^2 + 154.42x - 143.9\)

Are there any other ways to figure this out? Any other equations to get the same answer.

A Perfect square is when two rational numbers are multiplied together. That means no numbers such as 1.65, 5.89 etc.

sqrt(5) is a non perfect square as you get a irational number. 

2.236...

Do you see what i mean?

Agreed XD

Or let me say it in a better way This may help you with future problems 

A function is a set of input numbers that each have diffrent outputs  no to diffrent numbers will have the same output like maybe you have the numbers 1234 and 5 (input numbers) and you end up with 67890 then that = a function because no two outputs are the same.

Dont know if this helps.

A function Is a set of number that have diffrent outputs if a number has the same output but diffrent inputs then it is not a function this problem that is giving isn't a function because all outputs are five

True. Your calculations are correct!

In this problem the velocity changes as the marble accelerates towards the ground.

there is no A

the answer is K

\((x-4\pi)^2 + (y-1)^2 = 1\)

By rolling two revolutions the circle is displaced two circumferences to the right.

With radius 1 the circumference is 2pi.

The y coordinate of the center doesn't change when rolling horizontally.

To figure this out you need to have the speed which the ball is traveling.

V=s/t

Volocity equals speed over time.

Volocity is in meters per second

Look at the following graph :

The curve formed by a point on the circle as it rotates is known as a "cycloid"...the graph shows two rotations....the y coordinate for the center does not change....the new x value for the center is given by 4pi - sin (4pi)  = 4pi  So....the correct equation is

(x - 4pi)^2 + ( y - 1)^2 =  1

\(g = -9.8 ~m/s\\ h(t) = \dfrac{gt^2}{2} + v_0t+ h_0 = -4.9t^2 + 78.48\\ \text{at the end of it's fall the marble is at }h=0\\ 0=-4.9t^2 + 78.48\\ 4.9t^2 = 78.48\\ t^2 \approx 16.01632653\\ t \approx 4s\)

First one

-2 / ( x + 4)

Whatever x value makes the denominator 0 is a vertical asymptote....this is x = - 4

We hav e a lower degree polynomial  /  higher degree polynomial

Whenever this is the case, we will have a horizontal asymptote at y = 0

Second one

f(x) =  3 sqrt (x - 3)

The square root cannot be negative...so    x - 3 ≥ 0  ⇒    x ≥ 3

So....the domain is  [ 3, infinity )

The smallest y value occurs when  x = 0......this will lead to   3 sqrt (3 - 3) = 3sqrt (0)  = 0

And the function is aways increasing to infinity as x increases

So....the range is [ 0 , infinity )

Of course, we are gonna' need an app for this

I like this one :  https://keisan.casio.com/exec/system/14059932254941

The correct function is :  y=−11.41x^2+154.42x−143.9

It's really just a fancy version of the Pythagorean Theorem.....

See if this helps  : 

https://www.mathwarehouse.com/trigonometry/law-of-cosines-formula-examples.php

Thanks, PF.....I try   !!!!

Let's see StarStrike.....

Note that angle ABD = 180 - 35 =  145°

So angle ADB =   180 - 16 - 145 =  19°

So.....we can use the Law of Sines to find  thusly

AD / sin ABD = 950 / sin ADB

AD  / sin 145  =  950 / sin 19

AD =  950 * sin (145) / sin (19) ≈    1673.68

Now...ADC is a right triangle....so  we have that

sin 16   = DC / AD

sin 16 = DC / 1673.68

1673.68 * sin 16 =  DC  ≈ 461.33      and you are correct  !!!!

Thank you so much for your help! As always, your explanations are always easier to understand. :)

Same here.

First one

f(x) =   1 / (x - 3) + 1

g(x)  = 1 / (x + 4) + 3

g(x) is translated 7 units to the left and 2 units up

Second one

(x^2 + 3x - 28)  / ( x + 7)       factor the numerator

[ ( x + 7)   (x - 4) ]  /( x + 7)        cancel the ( x + 7)

x - 4

Since  x = - 7   makes the denominator of the original fraction = 0.....the "hole" occurs at x = -7

To find the y coordinate of the "hole"...put -7 into   x - 4    for x   =  - 7 - 4  =  -11

So....the "hole" occurs at  ( -7, -11)

So if im correct its 461.33m Tall.

2917.98 * 0.27563 = BD

BD = 804.303...

0.5735... * 804.303...=DC

DC= 461.33m Tall