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# If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc?

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If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 05:23
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If bc ≠ 0, what is the value of $$\frac{a^2 - b^2 - c^2}{bc}$$?

(1) |a| = 1, |b| = 2, |c| = 3
(2) a + b + c = 0
[Reveal] Spoiler: OA

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If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 05:47
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If bc ≠ 0, what is the value of $$\frac{a^2 - b^2 - c^2}{bc}$$?

The numerator of the expression will always equal -12

(1) |a| = 1, |b| = 2, |c| = 3
bc can equal 6 or -6.
The expression can equal 2 or -2
insufficient

(2) a + b + c = 0
Plugging in many different values and neg values the
Expression always equals 2
sufficient

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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 10:07
Bunuel:
According to second option: The value of expression will always be always be zero.
Right??
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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 10:39
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Expert's post
Hi Shree9975,

You have a theory, so you should try to prove that it's true. Try TESTing VALUES....

Choose values for A, B and C that "fit" the equation in Fact 2:

A+B+C = 0

Then plug those values into the question....Then choose a different set of values to TEST and plug those in....You don't need Bunuel to figure out if your theory is correct.

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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 11:44
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S1 cannot be sufficient, because we can make the denominator positive or negative without changing the value of the numerator.

For S2, if a+b+c = 0, then a = -b-c. So

\begin{align} \frac{a^2 - b^2 - c^2}{bc} &= \frac{(-b-c)^2 - b^2 - c^2}{bc} \\ &= \frac{(-1)^2(b + c)^2 - b^2 - c^2}{bc} \\ &= \frac{b^2 + 2bc + c^2 - b^2 - c^2}{bc} \\ &= \frac{2bc}{bc} \\ &= 2 \end{align}

Plugging in values seems needlessly time-consuming here.
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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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05 Jun 2015, 17:44
1
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If bc ? 0, what is the value of $$\frac{a^2 - b^2 - c^2}{bc}$$?

Stmt (1) |a| = 1, |b| = 2, |c| = 3

note: a^2, b^2, c^2 will be {+ve} , while b * c can be {+ve or -ve} 6

$$= \frac{1 - 4 - 9}{\pm 6} = \pm 2$$

insufficient

Stmt (2) a + b + c = 0

we know bc is in denominator ; lets write a in terms of (b+c)

$$a = -(b+c)$$ --- let's square both sides

$$a^2 = b^2 + c^2 + 2bc$$ --- plug in this value of a in main equation

$$\frac{a^2 - b^2 - c^2}{bc} = \frac{b^2 + 2bc + c^2 - b^2 - c^2}{bc} = \frac{2bc}{bc} = 2$$

sufficient

Ans: B
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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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08 Jun 2015, 02:57
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Expert's post
Bunuel wrote:
If bc ≠ 0, what is the value of $$\frac{a^2 - b^2 - c^2}{bc}$$?

(1) |a| = 1, |b| = 2, |c| = 3
(2) a + b + c = 0

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: The absolute value signs tell us that a = ±1, b = ±2, and c = ±3. In the numerator, each variable is squared, so their signs are irrelevant:

$$a^2 - b^2 - c^2 = 1^2 - 2^2 - 3^2 = 1 -4 - 9 = -12$$

However, while |b||c| = (2)(3) = 6, the denominator bc could be either 6 or –6, depending on the signs of b and c. Therefore:
$$\frac{a^2 - b^2 - c^2}{bc}=2$$ OR -2.

(2) SUFFICIENT: Given that a + b + c = 0, we know that a = - (b + c). Substitute this value of a into the expression in the question, and simplify, using one of the quadratic special products along the way:

$$\frac{a^2 - b^2 - c^2}{bc}=\frac{(-(b+c))^2 - b^2 - c^2}{bc}=$$
$$=\frac{(b+c)^2 - b^2 - c^2}{bc}=$$
$$=\frac{b^2+2bc+c^2 - b^2 - c^2}{bc}=$$
$$=\frac{2bc}{bc}=2$$

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Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc? [#permalink]

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25 Sep 2017, 19:54
(a-b-c)(a+b+c)+2bc / bc = 2
Re: If bc ≠ 0, what is the value of (a^2 - b^2 - c^2)/bc?   [#permalink] 25 Sep 2017, 19:54
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