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 0So, 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
Answer:
Phew!!!!