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# x(4 – x)

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x(4 – x) [#permalink]  15 Jun 2017, 08:18
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Question Stats:

50% (00:39) correct 50% (00:26) wrong based on 94 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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Re: x(4 – x) [#permalink]  19 Jun 2017, 10:10
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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 Dec 2017, 22:48
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I just simply plug number in,
when X is negative, X=-10
-10(4-(-10))=negative
when x is positive, x=10
10(4-10)= negative

B is always greater
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Re: x(4 – x) [#permalink]  19 Dec 2017, 23:23
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wongpcla wrote:
I just simply plug number in,
when X is negative, X=-10
-10(4-(-10))=negative
when x is positive, x=10
10(4-10)= negative

B is always greater

This time you got Lucky,
However in this type of Ques you have to consider every possible aspects. Since no information is given for x, so it can be negative, positive , fraction or integer
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Re: x(4 – x) [#permalink]  20 Dec 2017, 14:04
Expert's post
What panab01 pointed out is ONE of the most common mistakes made by the students.

Keep it in mind. Always.

Regards
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Re: x(4 – x) [#permalink]  20 Dec 2017, 17:58
If I plug four different number
(negative, positive , fraction or integer)
Can I avoid all the calculation?
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Re: x(4 – x) [#permalink]  21 Dec 2017, 16:37
Expert's post
wongpcla wrote:
If I plug four different number
(negative, positive , fraction or integer)
Can I avoid all the calculation?

I think no. At least to a minimum extent.

Regards
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Re: x(4 – x) [#permalink]  01 Mar 2018, 05:18
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In such questions, always play with the extremes. i.e
Plug in 0, -ve (betwen 0 and -1), -ve( less than -1), +ve(between 0 and 1) and +ve (greater than 1)

So in this question above, we need to solve in like an inequality expression as below:

Quantity A: x(4-x) or 4x - x^2
Quantity B: 6

We can add x^2 on both Quantities, the inequality will not effect, i.e

Quantity A: 4x
Quantity B: 6 + x^2

Similarly, we can add 6 on both Quantities, the inequality will not effect, i.e

Quantity A: 4x - 6
Quantity B: x^2

Now, if you put x = 0, 0.1, 10, -0.1 and -10

Actually, don't put -ve values, because in all negative values, Quantity B will be greater being positive.

Also, don't put 0, because Quantity A will be negative, while Quantity B remain 0.

Finally, only you need to put any +ve value, Quantity B will always greater.

So, Choice B is correct.

It was a good question.
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Re: x(4 – x) [#permalink]  28 Mar 2018, 11:06
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There is no limit for x.
If x is negative:
x(4-x) = 4x - x^2 is always negative because x^2 is always positive and thus -x^2 is negative, also 4x is negative.

If x is positive:
x(4-x) = 4x - x^2
For x = 4 the equation is 0. For 1 <= x <=3 it is positive and for values more than 4 for x, it is negative. Because x^2 is bigger for 4x.
x =1 4x-x^2 = 2
x =2 4x-x^2 = 4
x =3 4x-x^2 = 3
So either the equation is negative or is less than 6 ( 2,3,4). B is bigger than A.

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Re: x(4 – x) [#permalink]  19 Apr 2018, 05:18
We can find the maximum value of the expression x(4-x) by using the identity: Arithmetic Mean > Geometric Mean:
So,
AM > GM
(x + (4-x))/2 > \sqrt{x(4-x)}
2 > \sqrt{x(4-x)}
Squaring both sides
4 > x(4-x)
So, Quantity A would be less than 4. Hence Quantity B is larger

Another way of doing it is using a little bit of Calculus
The expression x(4-x) would be maximum/ minimum when we put first derivative of x(4-x) = 0
First Derivative: 4 - 2x = 0 --> x =2

So to see if the expression would have a maximum or minimum value at x =2, we see the second derivative of x(4-x). If the second derivative is negative, the expression x(4-x) would have the maximum value at 2 and if the second derivative is positive, the expression would have the minimum value at 2.
Second Derivative: -2

Hence the expression x(4-x) has the maximum value at 2 and that is 2(4-2) = 4
Re: x(4 – x)   [#permalink] 19 Apr 2018, 05:18
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