the two expressions have the same value when x = 5.
I can't see where you went wrong with your original equation, but the way I'd do it is:
Hence, area of shaded region:
Which is the original equation.
EDIT: Both your equation and the one in the question seem to work for the area, so maybe it's just a fault of the question.
Both formulas are correct (although since the problem said "show that the area is " the second method would be the "correct answer").
Skeeter pointed out that they both give the same result when x= 5 because, in fact, x must be 5!
I'm going to use "A" to represent the common value of AB and BC so that I can use "x" as a variable.
Set up a coordinate system with (0, 0) at B and the positive x axis extending from B through C. Then point A is at (0, A) and point C is at (A, 0). It is easy to see that the line AC is given by y= A- x.
Point E has coordinates (A-3, A- 2). Since it lies on line AC, we must have
A- 2= A- (A- 3)= 3 so A= 3+ 2= 5.
The "x" in the problem must be 5 and the two formulas give the same result.
While it is admittedly true in this case, I would argue that it is better to get into the habit of finding an identity rather than a formula, as it can then be applied to mathematically similar triangles rather than just that one. In short, yes, correct, but the identity is better mathematical practise in my opinion.