GYanggg

Hey, Rollingblade, we meet again!

Here is my solution from the same problem I posted.

We can designate \(\sqrt{a+x}=y\), which means that:

 \(\sqrt{a-y}=x\)

From this system, we can square both.

\(a+x=y^2\) and \(a-y=x^2\)

When we subtract the second equation from the first, we get:

\(x+y=y^2-x^2\)

This also is:

\(x^2-y^2+x+y=(x+y)(x-y+1)=0\)

From this, we have two cases:

Case 1: x + y = 0

In this case, y = -x and \(x^2-x-a=0\)

Case 2: y = x + 1 and \(x^2+x+1-a=0\)

Solving for x in both equations, we get.

\(x_{1,2}=\frac{1}{2}\pm\sqrt{a+\frac{1}{4}}\)

And

\(x_{3,4}=-\frac{1}{2}\pm\sqrt{a-\frac{3}{4}}\)

I hope this helped,

Gavin

Hi Ant. 

I just have a quick question, 

In the second part of the question, you said that the equilateral triangle has a side length of 10 and 9.

How can this be possible?

Hi SJ

We can plug in the values to find the recursive rule.

We plug in n = 1.

\(a_1=9(-5)^{1-1}=9\)

We imediately eliminate b and c.

We plug in n = 2

\(a_2=9(-5)^{2-1}=-45\)

We can see that only d works.

That is your final answer,

I hope this helped,

Gavin

We can first plug in the values of n, to find a pattern.

When n = 1, 2*1 - 1 = 1

When n = 2, 2*2 - 1 = 3

When n = 3, 2*3 - 1 = 5

...

When n = 15, 2*15 - 1 = 29

We see the pattern, 1, 3, 5, 7, ..., 29.

We find the sum 1 + 3 + 5 + ... + 29.

1 + 3 + ... n + 1 = n ^ 2

(29 - 1)^2 = 784

That is your final answer. 

I hope this helped,

Gavin.

You add up the fractions.

You can see the pattern in the denominators, 1, 2, 3, ....

Until p-1, where p is a fraction greater than 3. 

Would you like me to post the solution?

I hope this answered you question, 

Gavin.

Hey Guest!

Out of the 57 birds she caught again, 9 were tagged. 

Since she tagged 41 birds and only 9/57 were re-captured, we can do this to find the total number of birds:

\(41\cdot57\div9=\boxed{259.\overline{6}}\)

The final answer is D, 260.

I hope this helped,

Gavin

Case 1: 1 letter word

There are 4 ways to do this.

Case 2: 2 letter word

There are  \(4\cdot3\) , ways to do this.

Case 3: 3 letter word

There are \(4\cdot3\cdot2=24\), ways to do this.

Case 4: 4 letter word

There are \(4!=24\), ways to do this.

Add them all up, 4 + 12 + 24 + 24 = 64

That will be your final answer,

I hope this helped,

Gavin

Sorry, my previous answer was wrong.

Corner dominoes have two options, but regular squares have 3 or even 4.

Here is the correct one.

If the domino is oriented horizontally,

its position is determined by the leftmost square it covers,

for which there are \(7\cdot8=56\)

possibilities (the rightmost file is excluded).

Similar for vertical orientation. Thus, there are \(2\cdot56=112\) possibilities in total.

I hope this helped,

Gavin.

Happy Birthday IAmJeff!

I put in the functions in desmos, and here is what I got:

https://www.desmos.com/calculator/obzfylom2j

I split the trianlge into two, and calculated the summation of their areas.

(10 - 4)*2 + (10 - 4)*12 = 84

The area of the triangle is 84.

I hope this helped,

Gavin.