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 Post subject: GMAT Algebra
PostPosted: Thu May 02, 2013 1:07 pm 
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Joined: Tue Apr 13, 2010 8:48 am
Posts: 478
If y = (x – 5)² + (x + 1)² – 6, then y is least when x =

A. -2
B. -1
C. 0
D. 2
E. None of the above

(D) Let us transform the formula:
y = (x – 5)² + (x +1)² – 6 =
x² – 10x + 25 + x² + 2x + 1 – 6 =
2x² – 8x + 20 = 2 × (x² – 4x + 10) =
2 × ((x² – 4x + 4) + 6) =
2 × ((x – 2)² + 6)

Any square is greater or equal 0. Therefore the formula possess the least value when (x – 2)² = 0.
x – 2 = 0
x = 2
The correct answer is choice (D).
----------
Can you please explain how the +10 was broken down into the +4) + 6? I was able to get to 2 × (x² - 4x+10) but not any further.


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 Post subject: Re: GMAT Algebra
PostPosted: Thu May 02, 2013 1:07 pm 
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Quote:
+4) + 6?
This is the method to get (ax + b)² + c type of a formula.

Here we have x² – 4x + 10. Knowing the formula of a square of a sum, (a + b)² = a² + 2ab + b², we look at x² – 4x and look at it as x² – 2 × (2 × x). The missing b² must be 2² = 4.

Therefore we represent x² – 4x + 10 as (x² – 4x + 4) + 6.


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 Post subject: Re: GMAT Algebra
PostPosted: Thu Jul 03, 2014 5:46 am 
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Joined: Tue Apr 13, 2010 8:48 am
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Also, what's the rationale behind x = 2 and knowing that the 0 quantity will be the least quantity for y. Is that just a number line concept? I can't help but work out every answer choice just to be sure.

Thanks


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 Post subject: Re: GMAT Algebra
PostPosted: Thu Jul 03, 2014 5:53 am 
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The original formula for y is
y = (x – 5)² + (x + 1)² – 6

We transformed it into
y = 2 × ((x – 2)² + 6)
or
y = 2(x – 2)² + 12

In this form, we can say that y is not less than 12, because any square is non-negative.
y = non-negative number + 12

When it will be the least? – When that square is 0. This happens when x = 2.
-----------

Quote:
…and knowing that the 0 quantity will be the least quantity for y.
Note, that the main concept here is that any square is a non-negative number. So the least quantity for y is 12 (when the square is 0).


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