What is the value of x in the diagram?

5

Step-by-step explanation:

Pythagoras : a^2 + b^2 ≈ c^2

so, x^2 + (2x+2)^2 = (2x+3)^2

=> x^2 + 4x^2 + 8x + 4 = 4x^2 + 12x + 9

=> x^2 = 4x + 5

=> x^2 - 4x - 5 = 0

now solve the quadratic equating. a little bit of creative thinking and trying with the options of a (x-a)×(x+b)=0 approach (multiplying and adding of factors need to give us -5 but only -4x in the result, so we try a=-5 but then b=1 as "correcting" factor), and yes

(x-5)×(x+1) = x^2 - 4x - 5 = 0

this is only true, if at least one of the multiplication factors are 0.

so, x-5=0 or x+1=0. or both, but that is not possible.

x-5=0, when x=5.

x+1=0, when x=-1

-1 as the length of a side does not make any sense.

so, the only valid solution is x=5