The equation 24x^2 + 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? A) -16 B) -3 C) 3 D) 16
Answers
Answer:
Proficiency with algebra is an essential tool in understanding and being confident with mathematics. For those students who intend to study senior mathematics beyond the general level, factoring is an important skill that is frequently required for solving more difficult problems and in understanding mathematical concepts.
In arithmetic, finding the HCF or LCM of two numbers, which was used so often in working with fractions, percentages and ratios, involved knowing the factors of the numbers involved. Thus the factoring of numbers was very useful in solving a whole host of problems.
Similarly in algebra, factoring is a remarkably powerful tool, which is used at every level. It provides a standard method for solving quadratic equations as well, of course, as for simplifying complicated expressions. It is also useful when graphing functions.
Factoring (or factorising) is the opposite of expanding. Thus, using the distributive law,
3(x − 2) is the factored form of 3x − 6, and (x − 1)(x + 6) is the factored from x2 + 5x − 6.
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 B.