For and event i made 72 dessert plates, each containing the same number of pastries. I didn't need that many plates so i removed 12 plates, and redistributed the pastries, which meant that each remaining plate received two more pastries. How many pastries were on each plate to start with?

Work Shown:

x = starting number of pastries per plate

72x = total number of pastries

The total number of pastries is a multiple of 72.

If we take 12 plates away, then we have 72-12 = 60 plates left. Redistributing the pastries from those 12 plates, to the other 60, leaves us with no left over pastries (we get a remainder 0). This means that 72x must also be a multiple of 60 as well.

Let's find the LCM (lowest common multiple) of 60 and 72

First find the prime factorization of each:

60 = 2*2*3*5

72 = 2*2*2*3*3

We have the unique factors: 2, 3, 5

2 shows up at most 3 times, so 2^3 is one factor of the LCM

3 shows up at most 2 times, so 3^2 is a factor of the LCM

5 only shows up one time, so 5^1 is a factor of the LCM

The LCM is 2^3*3^2*5 = 8*9*5 = 360

Now we know that the total number of pastries must be a multiple of 360 in order to have 72x be a multiple of 60.

Divide 360 over 72 and 60 to get 360/72 = 5 and 360/60 = 6. We see that the number of pastries per plate has gone up by 1. We want it to go up by 2

Let's try the next multiple of 360

Divide 720 over 72 and 60 to get 720/72 = 10 and 720/60 = 12. The jump from 10 to 12 means we have two more pastries per plate. We found the answer.

Summary:

f(2x−x2)=−3x2+6x−1

step-by-step explanation:

set up the composite function and evaluate.

f(2x−x2)=−3x2+6x−1