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Thelegend2631
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If x and y are real numbers, the least possible value of the expressio 11 Jun 2021, 05:52
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38% (02:30) correct 62% (01:56) wrong based on 138 sessions

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If x and y are real numbers, the least possible value of the expression \(4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2\) .. is ?



(A) – 8
(B) – 4
(C) – 2
(D) 0
(E) 2
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If x and y are real numbers, the least possible value of the expressio 11 Jun 2021, 19:45
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\(4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2\)

\(4(x – 2)^2 – 2(x – 3)^2 + 4(y – 3)^2 \)

\(4x^2-16x+16-2x^2 +12x-18+ 4(y – 3)^2\)

\(2x^2-4x-2+ 4(y – 3)^2\)

\(2x^2-4x+2-4+ 4(y – 3)^2\)

\(2(x-1)^2 -4+ 4(y – 3)^2\)

\(2(x-1)^2 + 4(y – 3)^2 -4\)

Least possible value of the expression is -4, since minimum value square terms can take is 0.

B


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If x and y are real numbers, the least possible value of the expression \(4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2\) .. is ?



(A) – 8
(B) – 4
(C) – 2
(D) 0
(E) 2
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If x and y are real numbers, the least possible value of the expressio 11 Jun 2021, 07:18
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Not sure about Tacos but Kudos for sure for every solution..

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If x and y are real numbers, the least possible value of the expressio 11 Jun 2021, 20:58
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Asked: If x and y are real numbers, the least possible value of the expression \(4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2\) .. is ?

Least possible value of the expression is when : y = 3

At y = 3; \(E(x) = 4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2 = 4(x – 2)^2– 2(x – 3)^2\)
dE/dx = 8(x-2) - 4(x-3) = 0
4x -4 = 0
x = 1
Also d^2/dx^2 = 4 > 0

At x=1; E(x) = 4 - 2*4 = -4
Least value of the expression = - 4

IMO B
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If x and y are real numbers, the least possible value of the expressio 11 Jun 2021, 21:03
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Asked: If x and y are real numbers, the least possible value of the expression \(4(x – 2)^2 + 4(y – 3)^2 – 2(x – 3)^2\) .. is ?

Non calculus method to solve the question

At y=3; Minimum value of 4(y-3)^2 = 0
Now let us eliminate y from the expression.

\(E(x) = 4(x – 2)^2 – 2(x – 3)^2 = 2{(2x^2 - 8x +8) - (x^2 - 6x + 9)} = 2(x^2 -2x-1) = 2{(x-1)^2 -2} = 2(x-1)^2 - 4\)
E(x) is minimum when x = 1
Minimum value of the expression (at x=1) = -4

IMO B
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If x and y are real numbers, the least possible value of the expressio 12 Jun 2021, 01:10
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If given this question on the test, the first thing I would notice is that every answer choice is an integer ———> means we are looking for integer values for X and Y, more than likely


The first thing we can definitely do is eliminate the term of:

+ 4 (y - 3)^2

The output of a square will always be positive or 0 —— so if y = any other value than +3, we will be adding a positive value to the overall expression. We went to minimize, so make y = +3 so that this one expression drops out.

Minimize:

4 * (x - 2)^2 - 2 * (x - 3)^2


Whenever you subtract two numbers you can think of the subtraction as finding the distance on the number line between the 2 numbers being subtracted.

We also have the exponential effect of the squaring ——— we want X to be as far away from +3 as we can make it, without going so far as to allow the positive effect of +4 * (x - 2)^2 to overcome the magnitude of the negative value provided by -2 * (x - 3)^2


The first place to start is the following: make the distance from +2 —-> distance of zero so that the positive effect from +4 * (x - 2)^2 completely drops out

When X = +2

4 * (2 - 2)^2 - 2 * (2 - 3)^2 =

-2 * (-1)^2 =

-2

So we can eliminate two answers - D and E

As we move further left on the number line from +2, there will come a point at which the exponential effect of the squaring and *4 factor will lead to the positive magnitude of +4 (x - 2)^2 overcoming the negative magnitude of -2 * (x - 3)^2

When X = 1

4 * (-1)^2 - 2 * (-2)^2 =

4 - 8 =

-4

We can eliminate C as the minimization

Now all we need to decide is whether we can get the expression to equal = -8

When X = 0

4 * (-2)^2 - 2 * (-3)^2 =

16 - 18 =

-2

As we keep moving left on the number line, the positive value of +4 (x - 2)^2 will keep becoming relatively more and more larger in magnitude compared to -2 (x - 3)^2


Answer:
(B)

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If x and y are real numbers, the least possible value of the expressio 30 Aug 2025, 00:45
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