The sides of a triangle are x, x + 1, 2 x - 1 and its area is x root of 10. Find the value of x.

Ok, I'm going to start off saying there is probably an easier way of doing this that's right in front of my face, but I can't see it so I'm going to use Heron's formula, which is A=√[s (s-a) (s-b) (s-c) ] where A is the area, s is the semiperimeter (half of the perimeter), and a, b, and c are the side lengths.

Substitute the known values into the formula:

x√10=√{[ (x+x+1+2x-1) / 2][ ({x+x+1+2x-1}/2) - x][ ({x+x+1+2x-1}/2) - (x+1) ][ ({x+x+1+2x-1}/2) - (2x-1) ]}

Simplify:

x√10=√{[4x/2][ (4x/2) - x][ (4x/2) - (x+1) ][ (4x/2) - (2x-1) ]}

x√10=√[2x (2x-x) (2x-x-1) (2x-2x+1) ]

x√10=√[2x (x) (x-1) (1) ]

x√10=√[2x² (x-1) ]

x√10=√ (2x³-2x²)

10x²=2x³-2x²

2x³-12x²=0

2x² (x-6) = 0

2x²=0 or x-6=0

x=0 or x=6

Therefore, x=6 (you can't have a length of 0).