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# Which of the following is equivalent to for all values of x for which

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Re: Which of the following is equivalent to for all values of x for which [#permalink]
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$$\frac{2x^2(x + 3) -2x -6}{x^2 + 2x - 3}$$

= $$\frac{2x^2(x + 3) - 2 (x + 3)}{x^2 + 3x - x - 3}$$

= $$\frac{2(x^2 - 1)(x + 3)}{(x + 3)(x - 1)}$$

= $$2(x + 1)$$

= 2x + 2

Alternate approach: Put x = 0 in the given equation, we get the value as 2. Substitute x = 0 in the options, only option B remains.

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Bunuel wrote:
Which of the following is equivalent to for all values of x for which both $$\frac{2x^2(x + 3) -2x -6}{x^2 + 2x - 3}$$ expressions are defined?

(A) 2x^2 - 2
(B) 2x + 2
(C) x + 1
(D) 2x + 6
(E) x - 1

In this case, we can easily plug in values of x to test for equivalency.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we can also apply some algebraic factoring techniques to simplify the given expression.
Since I feel both techniques are equally fast, I'm going to test for equivalency since I'm less likely to make mistakes with that technique.

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

Let's evaluate the given expression for $$x = 0$$.

We get: $$\frac{2x^2(x + 3) -2x -6}{x^2 + 2x - 3}=\frac{2(0)^2(0 + 3) -2(0) -6}{(0)^2 + 2(0) - 3}=\frac{-6}{-3} = 2$$

We'll now evaluate each answer choice for $$x = 0$$ and eliminate those that don't evaluate to $$2$$
(A) $$2(0)^2 - 2 = -2$$. Doesn't evaluate to $$2$$. ELIMINATE.
(B) $$2(0) + 2 = 2$$. KEEP
(C) $$0 + 1 =1$$. Doesn't evaluate to $$2$$. ELIMINATE.
(D) $$2(0) + 6 = 6$$. Doesn't evaluate to $$2$$. ELIMINATE.
(E) $$0 - 1 = -1$$. Doesn't evaluate to $$2$$. ELIMINATE.

Originally posted by BrentGMATPrepNow on 28 Mar 2022, 13:59.
Last edited by BrentGMATPrepNow on 01 May 2022, 17:38, edited 1 time in total.
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Which of the following is equivalent to for all values of x for which [#permalink]
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BrentGMATPrepNow wrote:
We'll now evaluate each answer choice for $$x = 0$$ and eliminate those that don't evaluate to $$2$$
(A) $$2(1)^2 - 2 = 0$$. Doesn't evaluate to $$2$$. ELIMINATE.
(B) $$2(1) + 2 = 4$$. Doesn't evaluate to $$2$$. ELIMINATE.
(C) $$1 + 1 =2$$. KEEP
(D) $$2(1) + 6 = 8$$. Doesn't evaluate to $$2$$. ELIMINATE.
(E) $$1 - 1 = 0$$. Doesn't evaluate to $$2$$. ELIMINATE.

Brent, I think you accidentally subbed in 1 instead of 0 and got the wrong answer of C.
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