1. y - x = 3 is equivalent to y = x +3. Now replace this little thing over there and you get that:
6x(x + 3) = x^2(x + 3) + 9(x + 3) or
6x^2 + 18 x = x^3 + 3x^2 + 9x + 27
x^3 - 3x^2 - 9x + 27 = 0.
These are my favorite type of questions since you have to be creative and notice the connections:
x^3 - 3x^2 - 9x + 27 = (x^3 - 9x) - (3x^2 - 27) = x(x^2 - 9) - 3(x^2 - 9). As you can see, x^2 - 9 is a common factor, so we get that:
(x - 3)(x^2 - 9) = 0. Now we go on and we get that, since x^2 - 9 = (x - 3)(x + 3), (x-3)(x + 3)(x - 3) = 0. This equation has two roots, 3 and -3, so we cannot say for sure which is the correct answer.
2. Again, get creative!
6xy = x^2y + 9y is equivalent to x^2y - 3xy - 3xy + 9y = xy(x - 3) - 3y (x - 3) = y(x - 3)(x - 3) = 0. Now, there are only to possible ways that this thing could equal 0, and that's if:
a. x - 3 = 0 or x = 3. This is impossible since x^3 < 0 and 3^3 = 27
b. y = 0.
If y = 0 , then xy = 0.
Answer B
