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Math Expert V
Joined: 02 Sep 2009
Posts: 59083

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20 00:00

Difficulty:   95% (hard)

Question Stats: 36% (02:17) correct 64% (02:26) wrong based on 137 sessions

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What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

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Joined: 20 Mar 2014
Posts: 2570
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)

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1
happyface101 wrote:
iwantstanford wrote:
How did you know to split the -39 into 3 parts: -12, -18, and -9?

Hi Experts - can you please clarify how to recognize that we need to split -39 to solve?

I'm not sure how to recognize that this is the approach. Also, how do you know what to split -39 into?

The trick with these questions that ask you about max/min of a second degree polynomial is that you will invariably get a value of the form (a$$\pm$$b) wherein, the minimum value of any square = 0. Thus try to create perfect squares out of the given polynomials.

Realize that $$(a\pm b)^2 = a^2 \pm 2*a*b + b^2$$.

Given expression: $$−3x^2+12x−2y^2−12y−39$$ ---> try to create perfect squares :

$$(-3x^2+12x) + (-2y^2-12y)-39 ---> -3 (x^2-4x) -2 (y^2+6y) -39 ----> 3 (x^2-4x+4-4) -2 (y^2+6y+9-9) -39$$

---> $$3 (x^2-4x+4-4) -2 (y^2+6y+9-9) -39$$ ---> $$-3[(x-2)^2-4] -2 [(y+3)^2-9]-39$$ ----> $$-3(x-2)^2+12 -2(y+3)^2+18-39$$

----> $$-3(x-2)^2+12 -2(y+3)^2+18-39$$ --->$$-3(x-2)^2-2(y+3)^2-9$$

As mentioned above, the minimum value of a perfect square = 0 --> to maximize the given expression,

you need to put $$(x-2)^2 =0$$ and$$(y+3)^2=0$$ ---> giving you 0+0-9=-9 as the maximum value.

Any other values of$$(x-3)^2$$ and $$(y+3)^2$$will give you a smaller negative value hence 'minimizing' instead of maximizing the expression.

Hope this helps.
Intern  Joined: 01 Sep 2018
Posts: 1
Location: India
GPA: 4
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1
If you are familiar with basic differentiation, it would be very quick.

Solution: Let the given equation be 1.

Step 1.Find differentiation of equ 1 wr.to "X" and equate to "0" (This is the standard rule for a differentiation in order to get maximum value)
Hence, -6x+12=0
therefore x=2

Step 2.Then differentiate the equ 1 wr.to "Y" and equate to "0"
-4y-12=0
therefore y=-3

Step 3.Submit both x=2 and y=-3 in the equ. 1 to get the maximum value.

Differentiation approach can save lot of time while determining the maximum value of an expression.
Intern  Joined: 17 Sep 2015
Posts: 12
GMAT 1: 760 Q50 V42 GPA: 3.65

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3
How did you know to split the -39 into 3 parts: -12, -18, and -9?
Current Student B
Joined: 23 Nov 2016
Posts: 70
Location: United States (MN)
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Bunuel wrote:
Official Solution:

What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

$$-3x^2 + 12x -2y^2 - 12y - 39=-3x^2 + 12x-12-2y^2 - 12y-18-9=$$

$$-3(x^2-4x+4)-2(y^2+6y+9)-9=$$

$$=-3(x-2)^2-2(y+3)^2-9$$.

So, we need to maximize the value of $$-3(x-2)^2-2(y+3)^2-9$$.

Since, the maximum value of $$-3(x-2)^2$$ and $$-2(y+3)^2$$ is zero, then the maximum value of the whole expression is $$0+0-9=-9$$.

Bunuel, I'm sure you know this already but for this question a very quick shortcut would be Calculus, although obviously this isn't officially tested on the GMAT. I wanted to ask if you have you ever come across a Max/Min question where Calculus is a bad idea? Just want to make sure I don't fall into a trap, get caught in a problem that eats up a ton of time, etc.
Manager  G
Joined: 06 Sep 2018
Posts: 172
Location: Pakistan
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1
i solved this question in following way:
to maximize this expression, we need to maximize terms containing 'x', i.e
$$−3x^2+12x$$
and we need to maximize terms containing 'y', i.e
$$−2y^2−12y$$
So,
$$−3x^2+12x$$ will be maximum when x is positive and $$x=2$$.
$$−2y^2−12y$$ will be maximum when y is negative and $$y=-3$$
so by plugging in $$x=2$$ and $$y=-3$$ in given expression we get $$-9$$ and that's the maximum value we can get from this expression.
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Math Expert V
Joined: 02 Sep 2009
Posts: 59083

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Official Solution:

What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

$$-3x^2 + 12x -2y^2 - 12y - 39=-3x^2 + 12x-12-2y^2 - 12y-18-9=$$

$$-3(x^2-4x+4)-2(y^2+6y+9)-9=$$

$$=-3(x-2)^2-2(y+3)^2-9$$.

So, we need to maximize the value of $$-3(x-2)^2-2(y+3)^2-9$$.

Since, the maximum value of $$-3(x-2)^2$$ and $$-2(y+3)^2$$ is zero, then the maximum value of the whole expression is $$0+0-9=-9$$.

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Intern  Joined: 05 Aug 2015
Posts: 38

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iwantstanford wrote:
How did you know to split the -39 into 3 parts: -12, -18, and -9?

Hi Experts - can you please clarify how to recognize that we need to split -39 to solve?

I'm not sure how to recognize that this is the approach. Also, how do you know what to split -39 into?
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Manager  B
Joined: 27 Aug 2014
Posts: 66

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Bunuel wrote:
Official Solution:

What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

$$-3x^2 + 12x -2y^2 - 12y - 39=-3x^2 + 12x-12-2y^2 - 12y-18-9=$$

$$-3(x^2-4x+4)-2(y^2+6y+9)-9=$$

$$=-3(x-2)^2-2(y+3)^2-9$$.

So, we need to maximize the value of $$-3(x-2)^2-2(y+3)^2-9$$.

Since, the maximum value of $$-3(x-2)^2$$ and $$-2(y+3)^2$$ is zero, then the maximum value of the whole expression is $$0+0-9=-9$$.

'Hi
Is there a quick way to find the split?
CEO  S
Joined: 20 Mar 2014
Posts: 2570
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
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sinhap07 wrote:
Bunuel wrote:
Official Solution:

What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

$$-3x^2 + 12x -2y^2 - 12y - 39=-3x^2 + 12x-12-2y^2 - 12y-18-9=$$

$$-3(x^2-4x+4)-2(y^2+6y+9)-9=$$

$$=-3(x-2)^2-2(y+3)^2-9$$.

So, we need to maximize the value of $$-3(x-2)^2-2(y+3)^2-9$$.

Since, the maximum value of $$-3(x-2)^2$$ and $$-2(y+3)^2$$ is zero, then the maximum value of the whole expression is $$0+0-9=-9$$.

'Hi
Is there a quick way to find the split?

No, there is no "quickest" way than the ones mentioned above. With practice, this should not take you more than 2 minutes.
Current Student B
Joined: 26 Jan 2016
Posts: 98
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I'm confused how you got to this

(−3x 2 +12x)+(−2y 2 −12y)−39−−−>−3(x 2 −4x)−2(y 2 +6y)−39−−−−>3(x 2 −4x+4−4)−2(y 2 +6y+9−9)−39
(−3x2+12x)+(−2y2−12y)−39−−−>−3(x2−4x)−2(y2+6y)−39−−−−>3(x2−4x+4−4)−2(y2+6y+9−9)−3

I see what you factoried out the 2 and the -2, althought I'm not sure why the 2X and -2y were't factored out.

I'm lose how we got from 3(x²-4x) to 3(x²-4x+4-4) (i understand that it is the same thing, but why break it down like that?)

Kind of lost.

Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 59083

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brooklyndude wrote:
Bunuel wrote:
Official Solution:

What is the maximum value of $$-3x^2 + 12x -2y^2 - 12y - 39$$ ?

A. $$-39$$
B. $$-9$$
C. $$0$$
D. $$9$$
E. $$39$$

$$-3x^2 + 12x -2y^2 - 12y - 39=-3x^2 + 12x-12-2y^2 - 12y-18-9=$$

$$-3(x^2-4x+4)-2(y^2+6y+9)-9=$$

$$=-3(x-2)^2-2(y+3)^2-9$$.

So, we need to maximize the value of $$-3(x-2)^2-2(y+3)^2-9$$.

Since, the maximum value of $$-3(x-2)^2$$ and $$-2(y+3)^2$$ is zero, then the maximum value of the whole expression is $$0+0-9=-9$$.

Bunuel, I'm sure you know this already but for this question a very quick shortcut would be Calculus, although obviously this isn't officially tested on the GMAT. I wanted to ask if you have you ever come across a Max/Min question where Calculus is a bad idea? Just want to make sure I don't fall into a trap, get caught in a problem that eats up a ton of time, etc.

You can solve GMAT questions without calculus. So, no need to study it for the purpose of the test. On the other hand if you know that part of marth it won't hurt.
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to create perfect square u need to add and subtract (-b/2a)^2 in the equation ax^2 + bx + c = 0
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# M28-10

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