Question

please say answer I will mark as brainliest correct answer​

Answer 1

Question:

Solve the following equation:

$\displaystyle{\sf\:\dfrac{1}{x\:-\:3}\:+\:\dfrac{1}{5x\:-\:2}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:\dfrac{10\:+\:8\:\sqrt{3}}{23}\:}}\sf\:\quad\:OR\:\quad\:\boxed{\red{\sf\:x\:=\:\dfrac{10\:-\:8\:\sqrt{3}}{23}\:}}}$

Step-by-step-explanation:

We have given an equation in variable x.

We have to find the value of x.

Now,

$\displaystyle{\sf\:\dfrac{1}{x\:-\:3}\:+\:\dfrac{1}{5x\:-\:2}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

$\displaystyle{\implies\sf\:\dfrac{5x\:-\:2\:+\:x\:-\:3}{(\:x\:-\:3\:)\:(\:5x\:-\:2\:)}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

$\displaystyle{\implies\sf\:\dfrac{5x\:+\:x\:-\:2\:-\:3}{x\:(\:5x\:-\:2\:)\:-\:3\:(\:5x\:-\:2\:)}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

$\displaystyle{\implies\sf\:\dfrac{6x\:-\:5}{5x^2\:-\:2x\:-\:15x\:+\:6}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

$\displaystyle{\implies\sf\:\dfrac{6x\:-\:5}{5x^2\:-\:17x\:+\:6}\:=\:\dfrac{1}{-\:3x\:-\:2}}$

$\displaystyle{\implies\sf\:(\:6x\:-\:5\:)\:(\:-\:3x\:-\:2\:)\:=\:5x^2\:-\:17x\:+\:6}$

$\displaystyle{\implies\sf\:6x\:(\:-\:3x\:-\:2\:)\:-\:5\:(\:-\:3x\:-\:2\:)\:=\:5x^2\:-\:17x\:+\:6}$

$\displaystyle{\implies\sf\:5x^2\:-\:17x\:+\:6\:=\:-\:18x^2\:-\:12x\:+\:15x\:+\:10}$

$\displaystyle{\implies\sf\:5x^2\:-\:17x\:+\:6\:+\:18x^2\:+\:12x\:-\:15x\:-\:10\:=\:0}$

$\displaystyle{\implies\sf\:5x^2\:+\:18x^2\:-\:17x\:+\:12x\:-\:15x\:+\:6\:-\:10\:=\:0}$

$\displaystyle{\implies\sf\:23x^2\:-\:5x\:-\:15x\:-\:4\:=\:0}$

$\displaystyle{\implies\sf\:23x^2\:-\:20x\:-\:4\:=\:0}$

Comparing with ax² + bx + c = 0, we get,

• a = 23
• b = - 20
• c = - 4

Now, by quadratic formula,

$\displaystyle{\boxed{\pink{\sf\:x\:=\:\dfrac{-\:b\:\pm\:\sqrt{b^2\:-\:4ac}}{2a}\:}}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{-\:(\:-\:20\:)\:\pm\:\sqrt{(\:-\:20\:)^2\:-\:4\:\times\:23\:\times\:(\:-\:4\:)}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{20\:\pm\:\sqrt{400\:-\:92\:\times\:(\:-\:4\:)}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{20\:\pm\:\sqrt{400\:+\:368}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{20\:\pm\:\sqrt{768}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{20\:\pm\:\sqrt{16\:\times\:16\:\times\:3}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{20\:\pm\:16\:\sqrt{3}}{2\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{\cancel{2}\:(\:10\:\pm\:8\:\sqrt{3}}{\cancel{2}\:\times\:23}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{10\:\pm\:8\:\sqrt{3}}{23}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:x\:=\:\dfrac{10\:+\:8\:\sqrt{3}}{23}\:}}}\sf\:\quad\:OR\:\quad\:\underline{\boxed{\red{\sf\:x\:=\:\dfrac{10\:-\:8\:\sqrt{3}}{23}\:}}}}$

Answer 2

Refer the given aattachment