Question

Challenge time!!
The equation $\bf\dfrac{24x^2+25x-47}{ax-2}$=-8x-3-$\bf\dfrac{53}{ax-2}$ is true for all values of x ≠ 2/a. Where ''a'' is constant. Then find the value of a?
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Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\rm\dfrac{24x^2+25x-47}{ax-2}=-8x-3-\dfrac{53}{ax-2}$

On taking LCM on RHS, we get

$\rm :\longmapsto\:\rm\dfrac{24x^2+25x-47}{ax-2}=\dfrac{( - 8x - 3)(ax - 2) - 53}{ax-2}$

$\rm :\longmapsto\:\rm24x^2+25x-47=( - 8x - 3)(ax - 2) - 53$

$\rm :\longmapsto\:\rm24x^2+25x-47 + 53=( - 8x - 3)(ax - 2)$

$\rm :\longmapsto\:\rm24x^2+25x + 6= - 8a {x}^{2} + 16x - 3ax + 6$

$\rm :\longmapsto\:\rm24x^2+25x= - 8a {x}^{2} + (16 - 3a)x$

On comparing the coefficients, we get

$\rm :\longmapsto\:24 = - 8a \: \: and \: \: 25 = 16 - 3a$

$\rm :\longmapsto\:a= - \dfrac{24}{8} \: \: and \: \: 25 - 16 = - 3a$

$\rm :\longmapsto\:a= - 3 \: \: and \: \: 9 = - 3a$

$\rm :\longmapsto\:a= - 3 \: \: and \: \: - 3 = a$

$\bf\implies \:a \: = \: - \: 3$

Answer 2

$\huge\underline{\sf{Solution-}}$

Given expression is

$</p><p>\rm :\longmapsto\:\rm\dfrac{24x^2+25x-47}{ax-2}=-8x-3-\dfrac{53}{ax-2}:⟼ </p><p>ax−2</p><p>24x </p><p>2</p><p> +25x−47</p><p></p><p> =−8x−3− </p><p>ax−2</p><p>53$

On taking LCM on RHS, we get

$</p><p>\rm :\longmapsto\:\rm\dfrac{24x^2+25x-47}{ax-2}=\dfrac{( - 8x - 3)(ax - 2) - 53}{ax-2}:⟼ </p><p>ax−2</p><p>24x </p><p>2</p><p> +25x−47</p><p></p><p> = </p><p>ax−2</p><p>(−8x−3)(ax−2)−53</p><p>$

$</p><p>\rm :\longmapsto\:\rm24x^2+25x-47=( - 8x - 3)(ax - 2) - 53:⟼24x </p><p>2</p><p> +25x−47=(−8x−3)(ax−2)−53$

$\rm :\longmapsto\:\rm24x^2+25x-47 + 53=( - 8x - 3)(ax - 2):⟼24x </p><p>2</p><p> +25x−47+53=(−8x−3)(ax−2)</p><p>$

$Number=65$

$\rm :\longmapsto\:\rm24x^2+25x= - 8a {x}^{2} + (16 - 3a)x:⟼24x </p><p>2</p><p> +25x=−8ax </p><p>2</p><p> +(16−3a)x</p><p>$

On comparing the coefficients, we get

$\rm :\longmapsto\:24 = - 8a \: \: and \: \: 25 = 16 - 3a:⟼24=−8aand25=16−3a$

$\rm :\longmapsto\:a= - \dfrac{24}{8} \: \: and \: \: 25 - 16 = - 3a:⟼a=− </p><p>8</p><p>24$

and 25 − 16= −3a

$</p><p>\rm :\longmapsto\:a= - 3 \: \: and \: \: 9 = - 3a:⟼a=−3and9=−3a$

$\rm :\longmapsto\:a= - 3 \: \: and \: \: - 3 = a:⟼a=−3and−3=a$

$\bf\implies \:a \: = \: - \: 3⟹a=−3$