Final answer:
The value of 'a' that satisfies the equation (24x^2 + 25x - 47)/(ax - 2) = -8x - 3 - 53/(ax - 2) is found by equating coefficients after clearing the denominator, leading to the result that a = -3.
Step-by-step explanation:
To find the value of a in the equation (24x^2 + 25x - 47)/(ax - 2) = -8x - 3 - 53/(ax - 2), first note that the left side of the equation is a rational expression and the right side is a combination of a linear term and a separate rational expression. We are given that the equation holds for all values of x excluding x = 2/a, which suggests a common denominator on both sides of the equation. Thus, if we clear the denominator, we should get a polynomial that holds for all x (excluding the excluded value, which would make the denominator zero).
The equation can be written as:
- 24x^2 + 25x - 47 = -8x(ax - 2) - 53
Expanding the right side and moving all terms to one side gives us a quadratic equation in standard form. We set the coefficients of the corresponding powers of x equal because polynomials are equal if their corresponding coefficients are equal.
Equating coefficients gives us two equations:
- 24 (from x^2 term) = -8a (coefficient of x^2 term on the other side)
- 25 (from x term) = -8 (coefficient of x term on the other side)
From the first equation, we get a = -3 and we ignore the second equation because it does not contain a. The exclusion of the x term matching is actually a check to ensure our provided value of a does not create extra terms in the equation.
Therefore, the value of a that satisfies the given equation is -3.