A good place to start is to set 
to y. That would mean we are looking for 
to be an integer. Clearly, 
, because if y were greater the part under the radical would be a negative, making the radical an imaginary number, not an integer. Also note that since 
is a radical, it only outputs values from ![[0,\infty]](https://tex.z-dn.net/?f=%20%5B0%2C%5Cinfty%5D%20)
, which means y is on the closed interval: ![[0,120]](https://tex.z-dn.net/?f=%20%5B0%2C120%5D%20)
.
With that, we don't really have to consider y anymore, since we know the interval that is on.
Now, we don't even have to find the x values. Note that only 11 perfect squares lie on the interval , which means there are at most 11 numbers that x can be which make the radical an integer. All of the perfect squares are easily constructed. We can say that if k is an arbitrary integer between 0 and 11 then:
Which is strictly positive so we know for sure that all 11 numbers on the closed interval will yield a valid x that makes the radical an integer.
So,
The probability of drawing a yellow marble is
.
Since he replaces the marble after it is drawn out, the probability of drawing another yellow is the same:
.
The probability of drawing a red marble is
.
In order to get the probability of drawing 2 yellow marbles and 1 red marble, given that each marble is replaced after it is drawn out, we must multiply the fractions together.
Because the order of the marbles doesn't matter, we will multiply the probability by 3.
Step-by-step explanation below
the student is wrong because the Constant coefficient is 5 not 3.
constant coefficient is that which have no variable but in the opinion of the student the coefficient is 3 which have a variable x.
so the student is wrong.
Answer:
The parabola has an absolute minimum and its vertex is located at (1, 7).
Step-by-step explanation:
Since the directrix is below the focus, we infer that parabola has an absolute minimum, where there is a vertex, which is the midpoint of the line segment between (1, 10) and (1, 4). By definition of midpoint, we conclude that vertex is located at (1, 7).
