soumya1989 wrote:
What is the maximum value of \(-3x^2 + 12x -2y^2 - 12y -39\) ?
A. -39
B. -9
C. 0
D. 9
E. 39
This question requires us to perform a technique called
completing the square, which (in my opinion) is skirting the limits of what the GRE tests
That said here's the solution....
Take: \(-3x^2 + 12x -2y^2 - 12y -39\)
Rewrite as follows: \(-3(x^2 + 4x) - 2(y^2 + 6y) -39\)
Aside: At this point we need to recognize that \(x^2 + 4x\) will become a perfect square if we add 4, because \(x^2 + 4x + 4 = (x+2)(x+2) = (x+2)^2\)
Similarly \(y^2 + 6y + 9 = (y + 3)(y + 3) = (y + 3)^2\)
So I'm going to add the following to the algebraic expression: \(-3(x^2 + 4x + 4 - 4) - 2(y^2 + 6y + 9 - 9) -39\)
All I've done here is add and subtract 4 from the first quantity, and add and subtract 9 from the second quantity. Since the net effect is simply adding 0 the both quantities, this is a legitimate stepNow remove the \(-4\) and the \(-9\) from both brackets (they're not needed anymore)
Keep in mind that removing \(-4\) from the first set of brackets requires us to first multiply \(-4\) by \(-3\), removing \(-9\) from the second set of brackets requires us to first multiply \(-9\) by \(-2\)
When we do this we get: \(-3(x^2 + 4x + 4) + 12 - 2(y^2 + 6y + 9) + 18 -39\)
Simplify: \(-3(x^2 + 4x + 4) - 2(y^2 + 6y + 9) - 9\)
Factor: \(-3(x+2)(x+2) - 2(y+3)(y+3) - 9\)
Rewrite as follows: \(-3(x+2)^2 - 2(y+3)^2 - 9\)
At this point we need to recognize that \(-3(x+2)^2\) will be
less than or equal to 0 for all values of x So, the greatest the value of \(-3(x+2)^2\) is ZERO, and this occurs when \(x = -2\)
Likewise,\(-2(y+3)^2\) will be
less than or equal to 0 for all values of y So, the greatest the value of \(-2(y+3)^2\) is ZERO, and this occurs when \(y = -3\)
So, the greatest value of \(-3(x+2)^2 - 2(y+3)^2 - 9\) occurs when \(x = -2\) and \(y = -3\)
When we plug \(x = -2\) and \(y = -3\) into the equation, we get \(-9\)
So the greatest possible value is \(-9\)
Answer: B
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Brent Hanneson - founder of Greenlight Test PrepSign up for our GRE Question of the Day emails