sandy wrote:
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) \(5x^2 - 6x + 10\)
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) =
35So 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
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Brent Hanneson - founder of Greenlight Test Prep