Can someone explain to me how you use a quadratic equation (let x problems) in a pythagorean theorem type way?

Answer 1

OK. Suppose you have a right triangle whose sides are consecutive even integers. They would be x, x+2, and x+4.
x+4 would be the hypotenuse since it's longest.

So x^2 + (x+2)^2 = (x+4)^2

x^2 + x^2 + 4x + 4 = x^2 + 8x + 16 after doing out the FOIL

Move all to left making it = 0:

x^2 - 4x - 12 = 0

(x-6)(x+2) = 0

So x is either 6 or -2 but x can't be negative so cross the -2 out

Your sides are 6, 8, and 10

Answer 2

Those are mostly just for practice. Suppose the sides of a right triangle are x-1, 2x, and 2x+1.

Then (x-1)² + (2x)² = (2x+1)², and you multiply out the squares to get

x² - 2x + 1 + 4x² = 4x² + 4x + 1
x² - 6x = 0
x(x-6) = 0
x = 0, which we can't use because then the short side would be -1, or
x = 6, which makes the sides 5, 12, and 13.