If \(y = (x-5)^2+(x+1)^2 - 6\), then y is the Least when x GMAT Problem Solving

Question: If \(y = (x-5)^2+(x+1)^2 - 6\), then y is the least when x =

1. -2
2. -1
3. 0
4. 2
5. None of the above

Answer:

Solution with Explanation:

Approach Solution (1):

We are given that: \(y = (x-5)^2+(x+1)^2 - 6\)

Now let us transform the formula:

\(x^2-10x+25+x^2+2x+1-6\)
\(2x^2-8x+20=2 *(x^2-4x+10)\)
\(2*(x^2-4x+4+6)\)
\(2*((x-2)^2+6)\)

Any square is greater or equal to 0. Therefore the formula possess the least value when

\((x-2)^2=0\)

x – 2 = 0
x = 2

Correct Option: D

Approach Solution (2):

\(y=(x-5)^2+(x+1)^2-6=2x^2-8x+20=2x^2-8x+8+12\)

Thus \(y=2*(x^2-4x+4+12)=2*((x-2)^2+12)\)

Now, any square of the form \((a-b)^2\)will be minimum, when a = b and the minimum value = 0 (as for any square, \(x^2 \geq 0\) ,thus minimum value = 0).

Based on the discussion above:

Minimum value of y = \(2*((x-2)^2+12)\), will be when x = 2 and that value of x, y = 12

Correct Option: D

“If, then y is the least when x =?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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