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Math Expert V
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Question Stats: 73% (01:31) correct 27% (02:02) wrong based on 285 sessions

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Is $$abc = 0$$?

(1) $$a^2 = 2a$$

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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Math Expert V
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Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

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Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

I DISAGREE WITH B........2 actually reduces to b*(a+b)^2=0 which gives us b=0 or a=-b...(we cannot simply strike of (a+b)^2 in numerator and denominator since we do not know that a+b is not equal to 0.).

so it should be E.
Math Expert V
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suhasreddy wrote:
Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

I DISAGREE WITH B........2 actually reduces to b*(a+b)^2=0 which gives us b=0 or a=-b...(we cannot simply strike of (a+b)^2 in numerator and denominator since we do not know that a+b is not equal to 0.).

so it should be E.

That's not true. If a+b were 0, then $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$ would not be correct for any real a, b, and c.

When having something like x^2/x=y you can ALWAYS reduce by x and write x = y.

The answer is B.
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Bunuel wrote:
Official Solution:

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

Can someone explain the second step in this simplification process. It looks like factoring out the \sqrt{c}, but somehow \sqrt{c} became c. Is there some rule that allows this?
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The square root of c needs to be squared too
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I guess what I'm saying is that I've never seen this operation before. I believe it, because I basically get the same answer the long way:

(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)^2 = (a$$\sqrt{c}$$ + b$$\sqrt{c}$$)(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)
...and then the foil method

If it's a valid shortcut then I wonder if there is a proof or something for it so I know when I can use it in other situations.
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meshackb wrote:
I guess what I'm saying is that I've never seen this operation before. I believe it, because I basically get the same answer the long way:

(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)^2 = (a$$\sqrt{c}$$ + b$$\sqrt{c}$$)(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)
...and then the foil method

If it's a valid shortcut then I wonder if there is a proof or something for it so I know when I can use it in other situations.

This is like the example below:

(2*5+3*5)^2 = 5^2 * (2+3) = 25*5 = 125

The way you are mentioning the equation is also correct but instead of the traditional foil method , try to see it this way:

(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)^2 = (a$$\sqrt{c}$$ + b$$\sqrt{c}$$)(a$$\sqrt{c}$$ + b$$\sqrt{c}$$)

Take sqroot(c) common from both the terms on RHS of the equation above. sqroot (c)*sqroot (c) = c as sqroot (any number)= number ^0.5 (and x^p * x^q = x^ (p+q)).

So after doing this you get, sqroot (c) * sqroot (c) * (a+b)^2 = c * (a+b)^2. This is what Bunuel has done.

Originally posted by ENGRTOMBA2018 on 27 May 2015, 18:18.
Last edited by ENGRTOMBA2018 on 10 Jun 2015, 19:02, edited 1 time in total.
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Thanks, that clears it up!
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B
(1) Gives a= 2,0 Insuff
(2) Gives b = 0 or a=-b but a cannot be = -b as then equation's value will become undefined
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Bunuel wrote:
Is $$abc = 0$$?

(1) $$a^2 = 2a$$

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

Why in (1) $$a^2 = 2a$$, I cannot divide both sides by 'a' and get $$a = 2$$?
$$a^2 = 2a$$ -> $$(1/a) * a^2 = 2a * (1/a)$$ -> $$a=2$$

Is it wrong to do this way?

Edit: Answering my own question with a passage from Manhattan's Guide Book #2:
"The GMAT will often attempt to disguise quadratic equations by putting them in forms that do not quite look like the traditional form o f $$ax^2+ bx - c = 0$$."
Math Expert V
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guireif wrote:
Bunuel wrote:
Is $$abc = 0$$?

(1) $$a^2 = 2a$$

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

Why in (1) $$a^2 = 2a$$, I cannot divide both sides by 'a' and get $$a = 2$$?
$$a^2 = 2a$$ -> $$(1/a) * a^2 = 2a * (1/a)$$ -> $$a=2$$

Is it wrong to do this way?

Edit: Answering my own question with a passage from Manhattan's Guide Book #2:
"The GMAT will often attempt to disguise quadratic equations by putting them in forms that do not quite look like the traditional form o f $$ax^2+ bx - c = 0$$."

Yes, it's wrong. You cannot reduce this by a, because you'll loose a possible root a = 0. Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.
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Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

Bunuel,

In the way that you reduced the equation given in statement to. Should C also be squared. If I were to plug in 2 as A, 1 as B and 9 as C into that equation. I can’t get the equation to equal 81 unless the square root of C is ultimately squared.

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wildhorn wrote:
Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

Bunuel,

In the way that you reduced the equation given in statement to. Should C also be squared. If I were to plug in 2 as A, 1 as B and 9 as C into that equation. I can’t get the equation to equal 81 unless the square root of C is ultimately squared.

Posted from my mobile device

Not sure I can follow you...

Anyway, we are factoring $$\sqrt{c}$$ from the numerator in $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$:

$$b= \frac{(\sqrt{c})^2*(a*+b)^2}{a^2+2ab+b^2} - c$$;

$$b= \frac{c*(a+b)^2}{(a+b)^2} - c$$.
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Bunuel wrote:
wildhorn wrote:
Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

Bunuel,

In the way that you reduced the equation given in statement to. Should C also be squared. If I were to plug in 2 as A, 1 as B and 9 as C into that equation. I can’t get the equation to equal 81 unless the square root of C is ultimately squared.

Posted from my mobile device

Not sure I can follow you...

Anyway, we are factoring $$\sqrt{c}$$ from the numerator in $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$:

$$b= \frac{(\sqrt{c})^2*(a*+b)^2}{a^2+2ab+b^2} - c$$;

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

Bunuel,

thank you for the explanation. I see my mistake now. I have a follow up question. Would there be a simple way to reduce the numerator if there wasn't a constant multiple being used? For example, if the numerator was (A*C^1/2+B*D^1/2)^2 rather than (A*C^1/2+B*C^1/2)^2
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Okay from my personal experience, whenever you get a DS question, solve the Second Statement first.

In the above question . solving the Second statment first -

Taking \sqrt{c} common from the RHS , we get
$$b =( \sqrt{c}^2 *(a+b)^2)/(a+b)^2 - c$$
=> b = $$c - c$$
=> b = 0.

Hence we see that b = 0. Therefore the product abc will be 0. We can conclude from this statement that we dont even require the 1st Statement anymore.

Therefore Option B (Statement 2 ALONE is sufficient to solve the question).

Kindly Give Kudos if you liked my explanation.
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Bunuel wrote:
Official Solution:

In order $$abc = 0$$ to be true at least one of the unknowns must be zero.

(1) $$a^2 = 2a$$. Rearrange: $$a^2-2a=0$$. Factor out $$a$$: $$a(a-2)=0$$. Either $$a=0$$ or $$a=2$$. If $$a=0$$ then the answer is YES but if $$a=2$$ then $$abc$$ may not be equal to zero (for example consider: $$a=2$$, $$b=3$$ and $$c=4$$). Not sufficient.

(2) $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

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

$$b=c-c$$;

$$b=0$$. Sufficient.

I'm getting confused on a particular step, when $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$ is being simplified to $$b= \frac{c*(a+b)^2}{(a+b)^2} - c$$.
When solving it I'm getting $$b= \frac{c*(a^2+b^2)}{(a+b)^2} - c$$. And I'm not sure how $$(a^2+b^2)$$ is being written as $$(a+b)^2$$, as $$(a+b)^2 = {a^2+2ab+b^2}$$
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banerjr1 wrote:

I'm getting confused on a particular step, when $$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$ is being simplified to $$b= \frac{c*(a+b)^2}{(a+b)^2} - c$$.
When solving it I'm getting $$b= \frac{c*(a^2+b^2)}{(a+b)^2} - c$$. And I'm not sure how $$(a^2+b^2)$$ is being written as $$(a+b)^2$$, as $$(a+b)^2 = {a^2+2ab+b^2}$$

Let's do this with some intermediate steps so the math is more clear:

$$b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c$$

Factor out $$\sqrt{c}$$ from $$a*\sqrt{c}+b*\sqrt{c}$$

$$b= \frac{(\sqrt{c}*(a+b))^2}{a^2+2ab+b^2} - c$$

Distribute the square to $$\sqrt{c}$$ and $$a+b$$ (This is where you made your mistake — we can distribute a square to terms that are being multiplied together, but not terms that are being added together. This is why we get $$(a+b)^2$$ and not $$a^2 + b^2$$.)

$$b= \frac{\sqrt{c}^2*(a+b)^2}{a^2+2ab+b^2} - c$$

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

Either expand $$(a+b)^2$$ in numerator to $$a^2+2ab+b^2$$ OR factor $$a^2+2ab+b^2$$ in denominator to $$(a+b)^2$$

$$b= \frac{c*(a^2+2ab+b^2)}{a^2+2ab+b^2} - c$$ OR $$b= \frac{c*(a+b)^2}{(a+b)^2} - c$$

Cancel like terms from numerator and denominator (either $$a^2+2ab+b^2$$ or $$(a+b)^2$$)

$$b= c - c$$

$$b= 0$$

So you're totally correct that $$a^2 + b^2$$ doesn't equal $$(a+b)^2$$, but we should never get $$a^2 + b^2$$ based on exponent rules for distributing exponents.
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i am super confused,

Bunuel suppose a = (b* c)/b -c here as per this example we can write as a = c-c = 0

if a =(b* c)/b can we write as a = c by cancelling b rather if ab = bc can we cancel the b ???
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(1)
$$a^2 = 2a$$
$$a = 2$$

Don't know about any other value.
Not sufficient.

(2)
$$b = (a\sqrt{c} + b\sqrt{c})^2$$/$$(a + b)^2$$
$$b = {[a^2*c + b^2*c + 2abc]/[(a + b)^2]} - c$$
Taking C common from the numerator
$$b = {c*[a^2 + b^2 + 2ab]/[(a + b)^2]} - c$$
$$b = {c*[(a + b)^2]/[(a + b)^2]} - c$$
$$b = {c*[(a + b)^2]/[(a + b)^2]} - c$$
$$b = c - c$$
$$b = 0$$

if $$b = 0$$, then $$abc = 0$$. Hence, (2) is sufficient.

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~GMAC Re: M03-19   [#permalink] 12 Jul 2019, 22:27

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