Hello, Trentt!
Welcome aboard!
1/(1 - x) + 1/(1 + √x) .= .1/(1 - √x)
First, note that: .(1 - √x)(1 + √x) .= .1 - x
. . . . . . . . . .1 . . . . . 1 . . . . . . .1
We have: . ------ + --------- .= .--------
. . . . . . . . 1 - x . . 1 + √x . . . 1 - √x
Multiply through by (1 + √x)(1 - √x):
. . . . . 1 + 1 - √x . = . 1 + √x
And we have: . 2√x .= .1 . → . √x .= .½ . → . x .= .¼
Soroban multiplies through by (1+sqrt(x))(1-sqrt(x)) because:
1-x = (1+sqrt(x))(1-sqrt(x)),
so if you multiply both sides by this each of the terms leaves a simple
term in sqrt(x):
[1+sqrt(x))(1-sqrt(x)] [1/(1-x)]=(1-x)/(1-x) =1
[1+sqrt(x))(1-sqrt(x)] [1/(1+sqrt(x))]=(1-sqrt(x))
[1+sqrt(x))(1-sqrt(x)] [1/(1-sqrt(x))]=(1+sqrt(x))
so after multiplying through you are left with the equation:
1 + (1-sqrt(x)) = (1+sqrt(x)),
orL
2 sqrt(x) = 1
RonL