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# x(4 – x)

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x(4 – x) [#permalink]  15 Jun 2017, 08:18
Expert's post
00:00

Difficulty:

5% (low)

Question Stats:

0% (00:00) correct 0% (00:00) wrong based on 0 sessions

 Quantity A Quantity B x(4 – x) 6

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Joined: 10 Apr 2015
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Re: x(4 – x) [#permalink]  19 Jun 2017, 10:10
1
KUDOS
Expert's post
Carcass wrote:

 Quantity A Quantity B x(4 – x) 6

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

VERY tough question!! (165+)

I'm going to turn x(4 – x) into a perfect square
Before I do that, here are some other perfect squares:
x² + 6x + 9 = (x + 3)²
x² - 10x + 25 = (x - 5)²
x² - 4x + 4 = (x - 2)²
etc...

Given: x(4 – x) = 4x - x²
= -x² + 4x
= -1(x² - 4x)

What do we need to add to x² - 4x to make it a perfect square?
We need to add 4 to it to get x² - 4x + 4, which is equal to (x - 2)²
Of course, we can't just randomly add 4 to the given expression, since that totally changes the expression.
Instead, we're going to add 0 to the given expression. This is fine since adding 0 does not change anything.
HOWEVER, we're going to add 0 in a very SPECIAL WAY. We're going to add + 4 - 4 to the expression.
This is fine, since adding + 4 - 4 to the expression is the same as adding 0 to the expression.

We get: x(4 – x) = 4x - x²
= -x² + 4x
= -1(x² - 4x)
= -1(x² - 4x + 4 - 4)
= -1(x² - 4x + 4) + 4 [to remove -4 from the brackets, I had to multiply it by -1, since we are multiplying everything in the brackets by -1]
= -1(x - 2)² + 4

So, we can now write the following:
Quantity A: -1(x - 2)² + 4
Quantity B: 6

At this point, we must recognize that 4 is the GREATEST possible value of -1(x - 2)² + 4
We know this, because (x - 2)² is always greater than or equal to 0
So, -1(x - 2)² is always less than or equal to 0
So, the greatest value of -1(x - 2)² is 0. This occurs when x = 2
If 0 is the greatest possible value of -1(x - 2)², then 4 is the greatest possible value of -1(x - 2)² + 4

So, we get:
Quantity A: some number less than or equal to 4
Quantity B: 6

[Reveal] Spoiler:
B

Phew!!!!
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Re: x(4 – x)   [#permalink] 19 Jun 2017, 10:10
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