a 13- foot ladder is placed so that it reaches to a point on the wall that is 2 feet higher than twice the distance from the base of the wall to the base of the ladder. How high does the ladder reach?

Answer 1

x is the distance from base to wall

x^2 + (2x+2)^2 = 13^2

Now, you can expand that and solve for x, but the obvious hint is the 13' ladder. One of the common integer-sided triangles is the 5-12-13, where the lengths just exactly fit our problem.

The base is 5' from the wall, and the ladder top is 12' off the ground.