Solve for x. 2^x=4x. I really need the analytical solution.

From inspection one solution is x=4:

2^4 = 16 = 4 \times 4

I suggest you plot the functions y = 2^x and y=4x and you'll see that they actually cross at two points. One of those points is 4 as noted above. You won't be able to find a closed-form solution for the second point, but you can use an iteration technique to get a pretty good approximation of it.

Thank you for that

There was a very important business conference set-up in a Fortune 500 company's calender. On the d-day, a lot of pleasantries were exchanged. Everybody at the occasion shook hands with everybody else. All in all, there were 55,130,250 handshakes. How much less people would the conference have required to record only 31,988,001 handshakes?

First step: come up with a formula for how many handshakes there are between N people. If you think about how many handshakes there are if N=2, then N=-3, then N=4 you should be able tp come up with a general formula. Then use that formula to determine how many people there are for each of the two conditions.

First step: come up with a formula for how many handshakes there are between N people. If you think about how many handshakes there are if N=2, then N=-3, then N=4 you should be able tp come up with a general formula. Then use that formula to determine how many people there are for each of the two conditions.

I don't get it. Please help me to break it down. I'm not that good in word problem.

If there are N people in a room then each of those people must shake hands with (N-1) others. The number of ways two people can be selected out of N in a room is N(N-1). You then divide this by two because order is not important - if person A shakes person B's hand there is no need for person B to shake A's hand. So, the number of combinations of two people shaking hands is Combination(N,2) = N(N-1)/2. You can verify this by trying for N=2, 3, 4, etc. For example if N=3 the number of handsahakes according to the formula is 3x2/2 = 3. You can list those 3 quite easily: A-B, A-C, and B-C. I suggegst you try listing out all the hand shaking combinatins for four people and see if it matches the formula.

Now use this formula to determine the value for N that yields 55,130,250 handshakes.