Question

$\textbf{Question:}$
The Equation
$\frac{24 {x}^{2} + 25x - 47 }{ax - 2} = - 8x - 3 - \frac{53}{ax - 2}$is True for all values of
$x≠ \frac{2}{a} \:$ where a is a constant.

$\text{What is Value of a?}$
​

Answer 1

If 24x2+25x−47ax−2=−8x−3−53ax−2 is true for all values of ax≠2 and a is a constant, what is the value of a?

You seem to have missed something from the question! I think it should read ‘… is true for all values of x, except when …’

Let’s start by multiplying both sides of the equation by (ax−2):

24x2+25x−47=(ax−2)(−8x−3)−53

Adding 53 to both sides of the equation: 24x2+25x+6=(ax−2)(−8x−3)

Multiply out the terms in parentheses: 24x2+25x+6=−8ax2+16x−3ax+6

Subtracting (16x+6) from both sides of the equation: 24x2+9x=−8ax2−3ax

Adding 8ax2+3ax to both sides of the equation: 24x2+8ax2+9x+3ax=0

Factorising: 8(3+a)x2+3(3+a)x=0

For this to be true for a general value of x, we require the coefficients of x2 and x to be zero ⇒a=−3

Of course, as clearly pointed out in the question, we can’t have ax=2⇒−3x≠2⇒x≠−23

Answer: a = -3

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Answer 2

Answer:

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