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Quadratic Equation - Why is this the answer?

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by Poisson » Sun Jul 10, 2016 5:08 am
Please see the attached picture. I don't understand why crossing the respective numerators and denominators and multiplying the denominators doesn't work for this problem. For example, I know that a/b + c/d = (ad+bc)/bd. I tried using this method but did not get the correct answer. Please help me understand. Thank you!
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by Poisson » Sun Jul 10, 2016 5:11 am
Attached is a copy of my work. The answer is B but I got D and E using the shortcut for getting common denominators. Any help would be greatly appreciated.
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by [email protected] » Sun Jul 10, 2016 5:37 am
Hi Poisson,

If you take a good look at the first two 'lines' of your work, you'll see that you just removed the denominators from the calculation (and then you just combined the numerators) - that is NOT mathematically correct. If you want to approach this question algebraically, then that's your choice (although it will be a lot of work!), but you have to combine like terms before you 'cancel out' a fraction. You would likely find it MUCH faster to TEST VALUES (try TESTing A=2 or A=4 and track the results).

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by 800_or_bust » Sun Jul 10, 2016 5:49 am
Poisson wrote:Attached is a copy of my work. The answer is B but I got D and E using the shortcut for getting common denominators. Any help would be greatly appreciated.
You were good up to the point where you combined the fraction, but then you cancelled terms that couldn't be cancelled. And where did the 0 come from? That expression does not equal zero. It equals one of the answer choices.
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by 800_or_bust » Sun Jul 10, 2016 6:03 am
The correct way to approach this algebraically would be to factor out a -1 from the denominator of the second term as follows:

((a^2 + 12) / (a-3)) + (7a / (3-a)) = ((a^2 + 12) / (a-3)) + (7a / ((-1)(a-3))

Now the -1 can be freely moved to the numerator, so it simplifies as follows:

((a^2 + 12) / (a-3)) + (-7a / (a-3))

Now we have a common denominator and are able to add the numerators:

(a^2 - 7a + 12) / (a-3)

Factor the quadratic expression in the numerator:

(a-3)(a-4)/(a-3)

And cancel like terms.

The answer is a-4.
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by 800_or_bust » Sun Jul 10, 2016 6:09 am
Remember for all a & b, a - b = - (b - a). And (a-b)^2 = (b-a)^2, because squaring always produces a positive result and each of those expressions have the same absolute value.
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by GMATGuruNY » Sun Jul 10, 2016 9:57 am
If a=0, then (a²+12)/(a-3) + 7a/(3-a) = (0+12)/(0-3) + 0 = -4.
Now plug a=0 into the answer choices to see which yields a result of -4.
Only B works:
a-4 = 0-4 = -4.

The correct answer is B.
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