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07 Mar 2022, 06:45
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
89% (01:24) correct
11%
(02:06)
wrong
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Which of the following is equal to \((x - 2)^2 + (x - 1)^2 + x^2 + (x + 1)^2 + (x + 2)^2\)?
(A) \(5x^2\)
(B) \(5x^2 + 10\)
(C) \(x^2 + 10\)
(D) \(5x^2 + 6x + 10\)
(E) \(5x^2 - 6x + 10\)
(A) \(5x^2\)
(B) \(5x^2 + 10\)
(C) \(x^2 + 10\)
(D) \(5x^2 + 6x + 10\)
(E) \(5x^2 - 6x + 10\)
07 Mar 2022, 07:00
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Theory:
(x+y)^2 = x^2 + 2xy + y^2
(x-y)^2 = x^2 - 2xy + y^2
Lets look at our question. We know that the 2xy terms would cancel out for:
(x-2)^2 & (x+2)^2
(x-1)^2 & (x+1)^2
Therefore we would be left with (x^2+y^2) for each of the brackets:
(x^2 + 4) + (x^2 + 1)+ x^2 + (x^2 + 4) + (x^2 + 1) = 5x^2 +10
==> B
(x+y)^2 = x^2 + 2xy + y^2
(x-y)^2 = x^2 - 2xy + y^2
Lets look at our question. We know that the 2xy terms would cancel out for:
(x-2)^2 & (x+2)^2
(x-1)^2 & (x+1)^2
Therefore we would be left with (x^2+y^2) for each of the brackets:
(x^2 + 4) + (x^2 + 1)+ x^2 + (x^2 + 4) + (x^2 + 1) = 5x^2 +10
==> B
07 Mar 2022, 07:19
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Bunuel
Charli08 used the same rationale I would use to solve this question.
However, if you didn't see that shortcut, here's another technique that will help you avoid expanding and simplifying the original expression (which would be time-consuming).
Key concept: If two expressions are equivalent, they must evaluate to the same value for every possible value of x.
For example, since the expression 2x + 3x is equivalent to the expression 5x, the two expressions will evaluate to the same number for every value of x.
So, if x = 7, the expression 2x + 3x = 2(7) + 3(7) = 14 + 21 = 35, and the expression 5x = 5(7) = 35
So for example, if \(x = 1\), our original expression becomes: \((1 - 2)^2 + (1 - 1)^2 + 1^2 + (1 + 1)^2 + (1 + 2)^2\), which evaluates to be 15.
So, the correct answer must also evaluate to be 15, when \(x = 1\).
So, we'll plug \(x = 1\) into each answer choice....
(A) \(5(1)^2 = 5\). ELIMINATE
(B) \(5(1)^2 + 10 =\) 15 Perfect!
(C) \((1)^2 + 10 = 11\). ELIMINATE
(D) \(5(1)^2 + 6(1) + 10 = 21\). ELIMINATE
(E) \(5(1)^2 - 6(1) + 10 = 9\). ELIMINATE
Answer: B
