Question

Solve it:–
$\bf{ \dfrac{x - 3}{x + 3} \: + \: \dfrac{x + 3}{x - 3} \: = \: 2\dfrac{1}{2} }$

No spammed answers!
It's a quadratic equation.
Not linear equation.​

Answer 1

Information given to us:

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

What we have to calculate:

• We have to solve them and find out the values of x as it is a quadratic equation so we would be getting two answers.

Performing Calculations:

• Taking L.C.M.,

$: \longmapsto \: \sf{ \dfrac{(x - 3)(x - 3) \: + \: (x + 3)(x + 3)}{(x + 3)(x - 3)} \: = \: \dfrac{2 \times 2 + 1}{2} }$

• Opening brackets:

$: \longmapsto \: \sf{ \ \dfrac{x(x - 3) - 3(x - 3) \: + \: x(x + 3) +3(x + 3)}{x(x - 3) + 3(x - 3)} }$

$: \longmapsto \: \sf{ \ \dfrac{x \times (x - 3) - 3 \times (x - 3) \: + \: x \times (x + 3) +3 \times (x + 3)}{x \times (x - 3) + 3 \times (x - 3)} }$

$: \longmapsto \: \sf{ \dfrac{x {}^{2} - 3x + 9 + x {}^{2} + 3x + 3x + 9 }{x {}^{2} - 3x + 3x - 9} }$

• Solving it now,

$: \longmapsto \: \sf{ \dfrac{2x {}^{2} - 6x + 9 + 9 + 6x }{x {}^{2} - 9} }$

$: \longmapsto \: \sf{\dfrac{2x {}^{2} + 18}{x {}^{2} - 9 } \: = \: \dfrac{5}{2} }$

$: \longmapsto \: \sf{\dfrac{2(x {}^{2} + 9)}{x {}^{2} - 9 } \: = \: \dfrac{5}{2} }$

• Cross multiplying:

$: \longmapsto \: \sf{2 \times 2(x {}^{2} + 9) \: = \: 5(x {}^{2} - 9) }$

$: \longmapsto \: \sf{4(x {}^{2} + 9) \: = \: 5(x {}^{2} - 9) }$

$: \longmapsto \: \sf{4 \times (x {}^{2} + 9) \: = \: 5 \times (x {}^{2} - 9) }$

$: \longmapsto \: \sf{4x {}^{2} + 36 \: = \: 5x {}^{2} - 45 }$

• Reversing the sides,

$: \longmapsto \: \sf{5x {}^{2} - 4x {}^{2} \: - 45 - 36 \: = \: 0 }$

$: \longmapsto \: \sf{x {}^{2} - 81 \: = \: 0}$

• As it is a quadratic equation,

$: \longmapsto \: \sf{x \: = \: + \sqrt{81} }$

$: \longmapsto \: \boxed{\sf{x \: = \: + 9}}$

Now,

$: \longmapsto \: \sf{x \: = \: - \sqrt{81} }$

$: \longmapsto \: \boxed{\sf{x \: = \: - 9}}$

Conclusion:

• -9 and 9 is the answer

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Answer 2

Given :-

$\bf{ \dfrac{x - 3}{x + 3} \: + \: \dfrac{x + 3}{x - 3} \: = \: 2\dfrac{1}{2} }$

To Find:-

Value of x

Solution:-

Refer the attachment ✓