Question

Solve the following equation:
$\frac{(5-x)\sqrt{5-x}+(x-3)\sqrt{x-3}}{\sqrt{5-x}+\sqrt{x-3}}=2$

Answer 1

Consider the domains of function

$\sqrt{5-x}⟹x<5$

$\sqrt{x-3}⟹x>3$

By Cross Multiplication

$⟹(5-x)\sqrt{5-x}+(x-3)\sqrt{x-3}=2\sqrt{5-x}+2\sqrt{x-3}$

By Squaring on both sides

$(5-x{)}^3+(x-3{)}^3=4(5-x)+4(x-3)+8\sqrt{(5-x)(x-3)}$

$⟹3x^2-16x+45=4\sqrt{(}-x^2+8x-15)$

Squaring on both sides

$9x^4-96x^3+270x^2-1184x+2025=-16x^2+128x-240$

$⟹9x^4-96x^3+286x^2+1312x+2265=0$

which has no INTEGER solution