Question

Solve the following equations:

$\frac{(a-x)\sqrt{a-x}-(b-x)\sqrt{x-b}}{\sqrt{(a-x)}+\sqrt{(x-b)}}$ = a-b

Answer 1

The equation can be written as

$\frac{(a-x{)}^{3/2}+(x-b{)}^{3/2}}{(a-x{)}^{1/2}+(x-b{)}^{1/2}}$ = a-b

Form $\frac{p^3+q^3}{p+q}$ or $p^2+pq+q^2$

Whence we have

(a-x)-{a-x)${}^{1/2}$ (x- b)${}^{1/2}$ + (x-b) = a - b

or $\sqrt{(a-x)(x-b)}$ = 0 .

Hence x = a or x = b.