Question

find whether the equation 1/2x-3 +1/x-5=1 has real roots if real roots exist find them​

Answer 1

Equation

$\rm \frac{1}{2x - 3} + \frac{1}{x - 5} = 1 \\ \implies \rm \frac{x - 5 + 2x - 3}{(2x - 3)(x - 5)} = 1 \\ \implies \rm \: \frac{3x - 8}{2 {x }^{2} - 10x - 3x + 15 } = 1 \\ \implies \rm 3x - 8 = 2 {x}^{2} - 13x + 15 \\ \rm \implies \boxed{ 2{x}^{2} - 16x + 23 = 0}$

Discriminant

$\rm d = {b}^{2} \ - 4ac \\ \rm \implies \: d = 256 \ - 4 \times 2 \times 23 \\ \implies \: \rm \boxed{d = 72}$

Now

Using quadratic formula

$\rm x = \frac{ - b \pm \sqrt{d} }{2a} \\ \implies \rm \:x = \frac{16 \pm \sqrt{72} }{4} \\ \rm \implies \: x = \frac{16 \pm6 \sqrt{2} }{4} \\ \rm \implies \boxed{ x = \frac{8 \pm3 \sqrt{2} }{2} }$