Question

Solve for x: [ 1/(x -3) + 2/ (x - 2) ] = 8/x ; x≠ 0, 2,3​

Answer 1

Answer:

The value of x is 4 or 12 / 5.

Step-by-step-explanation:

The given equation is [ 1 / ( x - 3 ) + 2 / ( x - 2 ) ] = 8 / x.

We have to find the value of x where x ≠ 0, 2 or 3.

Now,

[ 1 / ( x - 3 ) + 2 / ( x - 2 ) ] = 8 / x

⇒ [ 1 × ( x - 2 ) + 2 × ( x - 3 ) / ( x - 3 ) × ( x - 2 ) ] = 8 / x

⇒ ( x - 2 + 2x - 6 / x² - 2x - 3x + 6 ) = 8 / x

⇒ ( x + 2x - 2 - 6 / x² - 5x + 6 ) = 8 / x

⇒ ( 3x - 8 / x² - 5x + 6 ) = 8 / x

⇒ ( 3x - 8 ) * x = 8 * ( x² - 5x + 6 )

⇒ 3x² - 8x = 8x² - 40x + 48

⇒ 8x² - 40x + 48 = 3x² - 8x

⇒ 8x² - 40x + 48 - 3x² + 8x = 0

⇒ 8x² - 3x² - 40x + 8x + 48 = 0

⇒ 5x² - 32x + 48 = 0

⇒ 5x² - 20x - 12x + 48 = 0

⇒ 5x ( x - 4 ) - 12 ( x - 4 ) = 0

⇒ ( x - 4 ) ( 5x - 12 ) = 0

⇒ x - 4 = 0 or 5x - 12 = 0

⇒ x = 4 or 5x = 12

⇒ x = 4 or x = 12 / 5

∴ The value of x is 4 or 12 / 5.

Answer 2

Step-by-step explanation:

$\sf \large \frac{x - 2 + 2 \bigg(x - 3 \bigg)}{ \bigg(x - 3 \bigg) \bigg(x - 2 \bigg)} = \frac{8}{x} \\ \\ \\ \\ \\ \\ \sf \large\frac{x - 2 + 2x - 6}{ {x}^{2} - 5x + 6 } = \frac{8}{x} \\ \\ \\ \\ \\ \\ \sf \large {3x}^{2} - 8x = {8x}^{2} - 40x + 48 \\ \\ \\ \\ \\ \\ \sf \large {5x}^{2} - 32x + 48 = 0 \\ \\ \\ \\ \\ \\ \sf \large \: \frac{x = 32 ± \sqrt{1024 - 960}}{10} \\ \\ \\ \\ \\ \\ \sf \large \frac{32 ± \sqrt{64} }{10} \\ \\ \\ \\ \\ \\ \sf \large \implies \underline{ \underline{ \boxed{ \sf \color{red} \large \frac{32 ±8}{10} = 4 , \frac{12}{5} }}}$