Here, I simplified it first to x^2 + 8x <= 16
then simplified it to x(x+8) <= 16
From this point I started choosing numbers.
when x = 1, 1(1+8) = 9
when x = 2, 2(1+8) = 20
So the max value of x has to be somewhere between 1 and 2 (not 2 for sure).
when x = - 9, -9(-9+8) = 1
when x = -10, -10(-10+8) = 20
So the min value of x has to be somewhere between -10 and -9 (not -10 for sure)
So if the max value is less than 2 and min value is greater than -10 for sure, the absolute difference between these two points on the number line has to be less than 8 (since absolute difference between 2 and -10 is 8).
So quantity A should actually be something lesser than -8, something like -8.something or lesser. Not sure of the exact value.
According to this reasoning quantity A is lesser than -8, but the answer to this question in the quiz is C.
Can someone help me understand where I’m going wrong?
Nope, r1 cannot equal 2 or 1. We know it is lesser than 2
and r2 cannot equal -10 or -9. We know it is greater than -10
Apart from that correction if we add both inequalities up, we should get
-9 < r1 + r2 < -7
So the max value of the sum of both the roots should be less then -7 and more than -9, that’s what we know for quantity A. It could be -8, sure, but it could also be -7.something or -8.something.
They’re (almost) logically equivalent for our case here. For example, 2 \leq 3 \leq 4 isn’t any different from 2 < 3 < 4 other than the latter being a stronger statement. Basically, a < b < c implies a \leq b \leq c but the reverse isn’t necessarily true. Anyhow, we can work with strict inequalities if that’s what you prefer.
This line of thought only gives us an interval to work with, which is insufficient in this case. If all values in the interval were less/more than -8 then we could’ve definitively answered the question as either A or B. However, since we’re aware that Quantity A assumes a single value in that interval, we can’t really compare it to another value (quantity B is technically another value in our interval) in that interval definitively. In particular, from our above working alone, we can’t say really distinguish between whether the answer should be C or D. As such, we move to other ideas more relevant to solving this problem because your approach isn’t conclusive.
As to how to solve the problem efficiently, you could invoke vieta’s formula for quadratics. You’re tasked with finding the sum of the roots of x^2 + 8x + 12 - 28, which is trivially -\frac{b}{a} = -8.
