Part a)
There are 52 letters (26 lowercase and 26 uppercase), 10 digits, and 6 symbols. There are 52+10+6 = 68 different characters to choose from.
- If there are 8 characters for this password, then we have 68^8 = 4.5716 * 10^14 different passwords possible.
- If there are 9 characters, then we have 68^9 = 3.1087 * 10^16 different passwords
- If there are 10 characters, then we have 68^10 = 2.1139 * 10^18 different passwords
- If there are 11 characters, then we have 68^11 = 1.4375 * 10^20 different passwords
- If there are 12 characters, then we have 68^12 = 9.7748 * 10^21 different passwords
Adding up those subtotals gives
68^8+68^9+68^10+68^11+68^12 = 9.9207 * 10^21
different passwords possible.
<h3>Answer: Approximately 9.9207 * 10^21 </h3>
======================================================
Part b)
Let's find the number of passwords where we don't have a special symbol
There are 52+10 = 62 different characters to pick from
- If there are 8 characters for this password, then we have 62^8 = 2.1834 * 10^14 different passwords possible.
- If there are 9 characters, then we have 62^9 = 1.3537 * 10^16 different passwords
- If there are 10 characters, then we have 62^10 = 8.3930 * 10^17 different passwords
- If there are 11 characters, then we have 62^11 = 5.2037 * 10^19 different passwords
- If there are 12 characters, then we have 62^12 = 3.2263 * 10^21 different passwords
Adding those subtotals gives
62^8+62^9+62^10+62^11+62^12 = 3.2792 * 10^21
different passwords where we do not have a special character. Subtract this from the answer in part a) above
( 9.9207 * 10^21) - (3.2792 * 10^21) = 6.6415 * 10^21
which represents the number of passwords where we have one or more character that is a special symbol. I'm using the idea that we either have a password with no symbols, or we have a password with at least one symbol. Adding up those two cases leads to the total number of passwords possible.
<h3>Answer: Approximately 6.6415 * 10^21</h3>
======================================================
Part c)
The answer from part a) was roughly 9.9207 * 10^21
It will take about 9.9207 * 10^21 nanoseconds to try every possible password from part a).
Divide 9.9207 * 10^21 over 1*10^9 to convert to seconds
(9.9207 * 10^21 )/(1*10^9) = 9,920,700,000,000
This number is 9.9 trillion roughly.
It will take about 9.9 trillion seconds to try every password, if you try a password per second.
------
To convert to hours, divide by 3600 and you should get
(9,920,700,000,000)/3600 = 2,755,750,000
So it will take about 2,755,750,000 hours to try all the passwords.
------
Divide by 24 to convert to days
(2,755,750,000)/24= 114,822,916.666667
which rounds to 114,822,917
So it will take roughly 114,822,917 days to try all the passwords.
------
Then divide that over 365 to convert to years
314,583.334246576
which rounds to 314,583
It will take roughly 314,583 years to try all the passwords
------------------------------
<h3>
Answers:</h3>
- 9.9 trillion seconds
- 2,755,750,000 hours
- 114,822,917 days
- 314,583 years
All values are approximate, and are roughly equivalent to one another.
. Now we can complete the square on the polynomial by taking half the linear term, squaring it, and adding it to both sides. Our linear term is 4. Half of 4 is 2, and 2 squared is 4, so we add 4 to both sides: 
and simplify to get 
. In this process we have created a perfect square binomial on the left, which happens to be 
. Now move the -6 back over by addition to get 
. The vertex is found at (-2, 6), the third choice down.