Question

x-1/x-2+x-3/x-4=3×1/3​

Answer 1

Answer:

$\large{\underline{\boxed{\bf x = 2 + \sqrt{2} \: \: or \: \: x = 2 - \sqrt{2}}}}$

Step-by-step explanation:

Given that-

$\sf \frac{x - 1}{x - 2} + \frac{x - 3}{x - 4} = 3 \times \frac{1}{3} \\ \\ \small \implies \sf \frac{(x - 1)(x - 4) + (x - 3)(x - 2)}{(x - 2)(x - 4)} = 1 \\ \\ \implies \sf \frac{(x {}^{2} - 4x - x + 4) + (x {}^{2} - 3x - 2x + 6)}{x {}^{2} - 4x - 2x + 8} = 1 \\ \\ \implies \sf \frac{x {}^{2} - 5x + 4 + x {}^{2} - 5x + 6 }{x {}^{2} - 6x + 8 } = 1 \\ \\ \implies \sf 2x {}^{2} - 10x + 10 = x {}^{2} - 6x + 8 \\ \\ \implies \sf 2x {}^{2} - 10x + 10 - x {}^{2} + 6x - 8 = 0 \\ \\ \implies \sf x {}^{2} - 4x + 2 =0$

Now, we got a quadratic equation,

x² - 4x + 2 = 0, here

• a = 1
• b = (-4)
• c = 2

Discriminant = b² - 4ac

⇒ D = (-4)² - 4 * 1 * 2

⇒ D = 16 - 8

⇒ D = 8

$\implies \tt x = \frac{ - b \pm \sqrt{D} }{2a} \\ \\ \implies \tt x = \frac{ - ( - 4) \pm \sqrt{8} }{2 \times 1} \\ \\ \implies \tt x = \frac{4 \pm 2 \sqrt{2} }{2} \\ \\ \implies \tt x = \frac{2(2 \pm \sqrt{2} )}{2} \\ \\ \implies \tt x =2 \pm \sqrt{2}$

Therefore,

$\boxed{ \bf x = 2 + \sqrt{2} \: \: or \: \: x = 2 - \sqrt{2} }$