Question

find the roots of quadratic equation

Answer 1

I hope this is correct.

Answer 2

Answer:

$\implies \: \red{\boxed{x \: = 5 \: , \: \frac{5}{2} }}$

Step-by-step explanation:

Solution :

According to the question,

$\small \red{ \: \frac{x - 1}{x - 2} + \frac{x - 3}{x - 4} = \frac{10}{3}} \\ \\ taking \: lcm \\ \\ \small \frac{(x - 1)(x - 4) + (x - 3)(x - 2)}{(x - 2)(x - 4)} = \frac{10}{3} \\ \\ \small \: \frac{ {x}^{2} - 5x + 4 + {x}^{2} - 5x + 6}{ {x}^{2} - 6x + 8} = \frac{10}{3} \\ \\ \small \: 3 \big(2 {x}^{2} - 10x + 10 \big) = 10 \big( \: {x}^{2} - 6x + 8 \big) \\ \\ \small \: 10 {x}^{2} - 6 {x}^{2} - 60x + 30x + 80 - 30 = 0 \\ \\ \: 4 {x}^{2} - 30x + 50 = 0 \\ \\ 2 {x}^{2} - 15x + 25 = 0 \\ \\ \: 2 {x}^{2} - 10x - 5x + 25 = 0 \\ \\ 2x(x - 5) - 5(x - 5) = 0 \\ \\ (x - 5)(2x - 5) = 0$

$\star \: case \: (1) \\ \implies \: (x - 5) = 0 \\ \\ \implies \: \boxed{x = 5} \\ \\ \star \: case \: (2) \\ \implies \: (2x - 5) = 0 \\ \\ \implies \: 2x = 5 \\ \\ \implies \: \boxed{ x = \frac{5}{2} }$