The monthly profit

ritumaheshwari02 wrote:The monthly profit, P(x), in dollars, from producing and selling x units of a particular product is given by P(x) = -(x - b)(x - c), where b and c are constants. If P(5) = P(95) = 0, what is the value of P(50) ?

A. $2,025
B. $2,475
C. $2,500
D. $6,525
E. $9,025

This problem gives us a great opportunity to look at something I often stress with my 1-on-1 students:

Mathematically Correct vs. GMAT Correct

What's the difference? Well everything that's GMAT Correct must, of course, be Mathematically Correct. But there are plentyÂ of Mathematically Correct ways to do this problem that are definitely GMAT Incorrect - that is, they take way longer than two minutes!

Let's look at just one of those GMAT Incorrect solutions:

First, FOIL out the formula for P(x):

P(x) = -(x-b)(x-c) = -(x^2-bx-cx+bc) = -x^2 + bx + cx - bc

Then, plug in 5 and 95 for x, setting each expression equal to zero (since P(5) = P(95) = 0).

-5^2 + 5b + 5c - bc = 0
-95^2 + 95b + 95c - bc = 0

Everything we have done so far is 100% Mathematically Correct. But look where it's gotten us: two terrible looking equations each with two variables! In theory, we could (maybe) go ahead and solve this. But it's going to be a nightmare and you'll never finish in two minutes. That's GMAT Incorrect.

So, how do we stay mathematically accurate, but get the job done in two minutes?

Here's one of my favorite pieces of mathematical advice:

A good mathematician is a lazy mathematician.

In the GMAT world, this doesn't mean be lazy with your studies! It means

On the GMAT, never do a computation you don't have to.

Let's approach this differently.

What we know:

P(x) = -(x-b)(x-c)
P(5) = 0
P(95) = 0

How can we make these facts work together? Plug in the given values for x, but think, don't compute (at least not yet).

P(5) = -(5-b)(5-c) = 0
P(95) = -(95-b)(95-c) = 0

In each equation, we have two parenthetical expressions that multiply together to zero. Therefore, in each case, one of the parentheticals must equal zero! In the first equation, this means that either b or c is equal to 5. In the second, either b or c is equal to 95. Does it matter which is which? Either way, we'll get the same P(x), so absolutely not!

We now have P(x) = -(x-5)(x-95) and we can just plug in to answer the question:

P(50) = -(50-5)(50-95) = -(45)(-45) = 45^2

This is basically the only computation we need to do. In real life, 45^2 is as easy as reaching for your smartphone. On the GMAT, spending too much time on these little computations can really add up, so let's not do anything unnecessary.

40^2 = 1600
50^2 = 2500

so 45^2 is in between. We're down to choices A and B already, but B is really close to 2500, so it's got to be A! This type of solution is exactly what I mean by *GMAT Correct*.

By the way, if you want to actually work it out exactly, here's a useful tip: FOIL! We mostly think of FOIL when doing algebra, but no one said we couldn't use this technique with numbers:

45^2 = 45*45 = (40+5)(40+5) = 1600 + 200 + 200 + 25 = 2,025

GMAT Pro Tip:

P(x) is a quadratic function given to us in factored form (meaning it's of the form a(x-b)(x-c) ... in this case a = -1). In this scenario, b and c are always the roots of the quadratic (meaning the x-values that give output = 0). When they tell us P(5) = 0 and P(95) = 0, that's the same as saying 5 and 95 are the roots, so with this bit of theory we get b = 5 and c = 95 without even thinking too hard.