If x is negative and y is positive, what is 4x2−4xy+y2

It is the square of (2x-y) or (y-2x) . Since y is positive and x is negative it should be (y-2x).

Ps You cannot take root of a negative number.

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1st, we can Factorize the Quadratic under the Square Root:

4(x)^2 - 4xy + (y)^2 =

(2x - y) (2x - y) =

sqrt( (2x - y)^2 )


Rule: the Square Root of a Variable Squared is equivalent to the Absolute Value of the Variable (logic: both functions yield the same result)

i.e——> sqrt( x^2 ) = [x]

Therefore, the Square Root of the Unknown Expression (2x - y)^2 SQUARED becomes:


The Absolute Value of —— [2x - y]


Lastly, we are Given that x = (-)Neg. and y = (+)Pos.

This means the Quantity Inside the Modulus will be a (-)Negative Input Expression


however, since the Output of a Modulus is always NON Negative—-> the Rule to Open the Modulus is the following:

When X < 0 ——— [X] = -(X)


Thus:

[2x - y] = -(2x - y) = -2x + y =


y - 2x

Answer -C-


Another way to think about it:

If X = (-)Neg. and Y = (+)

2x - y = (-)Negative Result

However, the Output of an Absolute Value Modulus can NEVER be (-)Negative. And the Answer Choices are giving us the result of an Absolute Value Expression.

Thus, 2x - y could never be the correct answer.

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Concept used
(a-b)^2 = a^2 + b^2 -2ab
lxl = \sqrt{x^2}

Given expression can be reduced to l2x-yl;
Now x is -ve and y is +ve= (y-2x) C

The common mistake being tested here is that we always forget that square root is always positive i.e. \(\sqrt{y^2}\)= |y|

Test x=-1 and y=1 to find 3
plug-in x=-1 and y=1 into each answer choice to find the correct answer.

The equation under the square root presented is: \(\sqrt{\ 4x^2+y^2-4xy}\)

\(\ 4x^2+y^2-4xy\) =

\(\left(2x-y\right)^2\ or\ \left(y-2x\right)^2\)

=\( \left|2x-y\right|\)

Since x is negative and y is positive.

\(\left|2x-y\right|\ =-\left(2x-y\right)\ =\ y-2x\)

I think this question can be solved by inputting the values.

Let x = -1 and y =1

So Value is (4* 1 + 4 +1 )^2 = 9^2 = 3

So Put the values in options to get 3

Option C gives 3

OA- C

\(\sqrt{4x^2−4xy+y^2}\)

= \((2x-y)^2 = |2x-y|\)

Since x is negative and y is positive = \(y - 2x\)

Answer is C.

The key to this question is to know that you can't take the root of a negative number.