Question

√x/x-3 + √x-3 =5/2


brainly sir please answer​

Answer 1

Hey bud,



I'm hoping your question looks something like this,

$\sqrt{ \frac{ x}{x - 3} } + \sqrt{x - 3} = \frac{5}{2} \\ \\ \\ = \frac{ \sqrt{x} }{ \sqrt{x - 3} } + \sqrt{ x - 3} = \frac{5}{2}$


Let's make a common denominator first,
$\frac{ \sqrt{x} + x - 3}{ \sqrt{x - 3} } = \frac{5}{2}$


squaring both sides gives,

$\frac{ {( \sqrt{x} )}^{2} + {(x - 3)}^{2} }{x - 3} = \frac{25}{4} \\ \\ \frac{x + {x}^{2} - 6x + 9}{x - 3} = \frac{25}{4} \\ \\ \frac{ {x}^{2} - 5x + 9 }{x - 3} = \frac{25}{4}$
Doing a bit of crossmultiplication gives,


$4 {x}^{2} - 20x + 36 = 25x - 75 \\ \\ 4 {x}^{2} - 45x + 111 = 0$


Solving this quadratic equation gives it's roots as,

$\frac{45 + \sqrt{249} }{8} \\ \\ \: and \\ \\ \frac{45 - \sqrt{249} }{8}$



And so these are the two possible answers to your x here.



Cheers.
Kudos to Brainly.

Answer 2

Answer:

x = 4 or x = -1

Step-by-step explanation:

$\sqrt{\dfrac{x}{x - 3} } + \sqrt{\dfrac{x - 3}{x} } = \dfrac{5}{2}$

Define y:

$\text{Let y} = \sqrt{\dfrac{x}{x - 3} }$ ,

$\text{Therefore} \sqrt{\dfrac{x - 3}{x } } = \dfrac{1}{y}$

Rewrite the equation:

$y + \dfrac{1}{y} = \dfrac{2}{5}$

Solve y:

$y + \dfrac{1}{y} = \dfrac{2}{5}$

$\dfrac{y^2 + 1}{y} = \dfrac{2}{5}$

$5(y^2 + 1) = 2y$

$5y^2 + 5 = 2y$

$5y^2 - 2y + 5 = 0$

$y = 2 \text{ or } y = \dfrac{1}{2}$

Solve x (if y = 2):

$\sqrt{\dfrac{x}{x - 3} } = 2$

$\dfrac{x}{x - 3} = 4$

$x = 4(x - 3)$

$x = 4x - 12$

$3x = 12$

$x = 4$

Solve x (if y = 1/2):

$\sqrt{\dfrac{x}{x - 3} } = \dfrac{1}{2}$

$\dfrac{x}{x - 3} } = \dfrac{1}{4}$

$4x = x - 3$

$3x = - 3$

$x = -1$