Question

The equation
24x2+25x−47/ax−2 =−8x−3−53/ax−2
is true for all values of x≠2/a
where a is a constant.

What is the value of a?

Answer 1

Answer:

a = −3

Step-by-step explanation:

[24x² + 25x - 47] /ax - 2 = - 8x - 3 - [53/ax-2]

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

24x² + 25x − 47 = (ax−2)(−8x−3)−53

=> 24x² + 25x + 6 = (ax−2)(−8x−3)

=> 24x² + 25x + 6 = −8ax² + 16x − 3ax + 6

=> 24x² + 9x = - 8ax² - 3ax

=> 24x² + 9x + 8ax² + 3ax = 0

=> 8x²(a + 3) + 3x(a+3) = 0

=> (a + 3)(8x² + 3x) = 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

Answer 2

Answer:

There are two ways to solve this question. The faster way is to multiply each side of the given equation by ax−2 (so you can get rid of the fraction). When you multiply each side by ax−2, you should have:

24x^2+25x−47=(−8x−3)(ax−2)−53

You should then multiply (−8x−3) and (ax−2) using FOIL.

24x^2+25x−47=−8ax^2−3ax+16x+6−53

Then, reduce on the right side of the equation

24x^2+25x−47=−8ax^2−3ax+16x−47

Since the coefficients of the x2-term have to be equal on both sides of the equation, −8a=24, or a=−3.

The other option which is longer and more tedious is to attempt to plug in all of the answer choices for a and see which answer choice makes both sides of the equation equal. Again, this is the longer option, and I do not recommend it for the actual SAT as it will waste too much time.

The final answer is -3