A10 foot ladder is placed against a house. if the base of the ladder is 3.5 feet from the base of the house, how high does the ladder reach on the house

Think of a right triangle whose hypotenuse is the ladder length, or 10 ft.  
We can apply the Pythagorean Theorem here:

10^2 = 3.5^2 + y^2, where y is the height of the top of the ladder as measured from the ground.

Then 100 = 12.25 + y^2, so that y^2 = 100 - 12.25, or y^2 = 87.75.

Then the answer is y = sqrt(87.75) = 9.37 ft.

The answer would be quadratic trinomial