Alan

This question was dealt with here:  https://web2.0calc.com/questions/reposted-because-the-answer-i-got-last-time-was-incorrect#r8 

"The square of an integer is 812 greater than the integer itself.  What is the sum of all integers for which this is true?"

Let n be the integer.

Then we have  n² - 812 = n  or  n² - n - 812 = 0

This factors as  (n + 28)(n - 29) = 0 

You can take it from here.

This question has appeared several times previously.  Search the answers of heureka or myself.

"What is the least positive integer multiple of 30 that can be written with only the digits 0 and 2?"

Must begin with 2 and end with 0.  The sum of the digits must be divisible by 3.  The smallest therefore seems to be 2220  (which is 74*30).

It must be an integer that is not even and not divisible by 3.  It must have a remainder of 2 when divided by 7 (because the set starts with 2);

I make 2011 the first to satisfy these conditions.

As follows:

I'll leave you to sum the two values.

Don't simply accept the results I've presented - check them yourself!

Speed is the rate of change of distance with respect to time, so the greatest speed will be at the point where the graph of distance against time is steepest.

It's the standard formula for solving quadratic equations (though I should have written "t = ..." not "x = ..."), which is often necessary when the quadratic equation doesn't factor nicely (as yours doesn't).

Use the quadratic equation to get the two values of t:

\(t = {-(-25) \pm \sqrt{(-25)^2-4\times 75} \over 2}\)

I'll leave you to crunch the numbers.

Edited to replace x by t.

"How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths a, b and c such that a, b, c is an arithmetic sequence?"