Question

How do you solve for x ? `(x-2)(x-3)=34/33^2`

Answer 1

`100/33 " and " 65/33`

Explanation:

`(x-2)(x-3) = 34/33^2`
`=> (33x-66)(33x-99) = 34`

being hopeful that `(33x-66) and (33x-99)` are integers , at this point, I note that this equation is basically a factorization of 34 such that `a.b=34` and `a-b = 33`

Naturally, the factors are 34 and 1 or (-1) and (-34).

There are two options :

Case I : `a=34 and b =1\ => x = 100/33`

Case II : `a=-1 and b = -34 \ => x= 65/33`

Answer 2

`x=65/33 or 100/33`

Explanation:

Let x-3 =a then x-2==a+1

`(x-2)(x-3)=(33+1)/33^2`

`=>(a+1)a=33/33^2+1/33^2`

`=>a^2+a-1/33^2-1/33=0`

`=>(a+1/33)(a-1/33)+1(a-1/33)=0`

`=>(a-1/33)(a+1/33+1)=0`

`=>(a-1/33)(a+34/33)=0`

`a=1/33 and a=-34/33`

when `a=1/33`

then `x-3=1/33`

`x=3+1/33=100/33`

when `a=-34/33`

then `x-3=-34/33`

`x=3-34/33=(99-34)/33=65/33`