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Intern  Joined: 24 Nov 2015
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What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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What is the value of x?

(1) $$x^2 + x + 10 = 16$$
(2) $$x = 4y^4+2y^2+2$$

Originally posted by GMAT4937 on 26 Jan 2016, 13:54.
Last edited by GMAT4937 on 26 Jan 2016, 14:15, edited 1 time in total.
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Quote:
What is the value of x ?

(1) x² + x + 10 = 16

(2) x = 4y⁴ + 2y² +2

Target question: What is the value of x?

Statement 1: x² + x + 10 = 16
Rewrite as: x² + x - 6 = 0
Factor: (x + 3)(x - 2) = 0
So, x = -3 OR x = 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x = 4y⁴ + 2y² + 2
As you can see, the value of x depends on the value of y.
Consider these two cases:
Case a: y = 0, in which case x = 4(0⁴) + 2(0²) + 2. Evaluate to see that x = 2
Case b: y = 1, in which case x = 4(1⁴) + 2(1²) + 2. Evaluate to see that x = 8
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x = -3 OR x = 2
Statement 2: Since y⁴ and 2y² are greater than or equal to zero FOR ANY value of y, we can see that 4y⁴ + 2y² + 2 MUST BE POSITIVE. In other words, x must be POSITIVE.

If x must be POSITIVE, then x MUST equal 2

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Intern  Joined: 24 Nov 2015
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What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Thanks for your explanation! I was thrown off by the extra variable "y" in the second statement, but when I got a positive and negative solution for x in statement one, that should've helped me figure out how to make use of the second.
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Hi,

Can anyone clarify my query here?

Nothing is stated about 'y' rite...??
So y can be irrational, rational, integer, etc.

If i assume an extreme case: y an irrational number, then the answer is E rite?

Please clarify? Should i not take this case into consideration?
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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2
HARRY113 wrote:
Hi,

Can anyone clarify my query here?

Nothing is stated about 'y' rite...??
So y can be irrational, rational, integer, etc.

If i assume an extreme case: y an irrational number, then the answer is E rite?

Please clarify? Should i not take this case into consideration?

For (2): regardless what real number y is (rational, irrational, ...), y in even power would be nonnegative. Thus, $$x = 4y^4+2y^2+2=nonnegative+nonnegative+positive=positive$$.

From (1) x = -3 OR x = 2 and from (2) that x is a positive number, therefore when we combine the statements, x can only be 2. Sufficient.

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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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1
1
This is good question to help you make start thinking on a higher level for the GMAT and start for DS questions start asking the question "Why are they giving me this information?"

The question is x = ?

Statement 1:

When factored works out to be x = 2 or x = -3. This is clearly insufficient as we have two answers for x, one positive and one negative. When I solve and quadratic and I get one positive and one negative answer its usually a good idea to double check the question to make it doesn't provide a small bit of info that could eliminate one of the answer choices. In this case, there's no additional info so we'll carry on.

Statement 2:

Just by looking at the question we can see it's insufficient. Plug in 1 for y and we get one answer and plug in 2 and we get a different answer.

Statements 1 & 2:

This is where it might look insufficient. But if we look back at statement 2 and ask ourselves why did the test makers give us this info? We have one negative answer for x and one positive answer. Is there any value for y we can plug in that will give us an negative answer for x? No, all of the powers of Y are even so the answer will always be positive. Therefore, X = 2 and we pick answer C.
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Re: What is the value of x?  [#permalink]

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souvonik2k wrote:
What is the value of x?

(1) x² + x + 10 = 16
(2) x = 4y⁴ + 2y² +2

Target question: What is the value of x?

Statement 1: x² + x + 10 = 16
Given: x² + x + 10 = 16
Subtract 16 from both sides: x² + x - 6 = 0
Factor: (x + 3)(x - 2) = 0
So, EITHER x = -3 OR x = 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x = 4y⁴ + 2y² +2
Since we don't know anything about the value of y, there are many possible values of x. Here are two:
Case a: if y = 0, then x = 4(0)⁴ + 2(0)² +2 = 2
Case b: if y = 1, then x = 4(1)⁴ + 2(1)² +2 = 8
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x = -3 OR x = 2
Statement 2 tells us that x is POSITIVE.
How do we know that x is POSITIVE?
Well, 4y⁴ ≥ 0 and 2y² ≥ 0 for all values of y. So, the SUM 4y⁴ + 2y² +2 MUST BE POSITIVE.
Since x = 4y⁴ + 2y² +2, we know that x is POSITIVE.
If x is positive, then x cannot equal -3, which means x must equal 2
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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GMAT4937 wrote:
What is the value of x?

(1) $$x^2 + x + 10 = 16$$
(2) $$x = 4y^4+2y^2+2$$

(1) $$x^2 + x + 10 = 16$$
$$x^2 + x + 10 - 16 = 0$$
$$x^2 + x + 6 = 0$$
$$x^2 + 3x - 2x + 6 = 0$$
$$x(x + 3) - 2 (x + 3) = 0$$
$$(x-2)(x + 3) =0$$

Therefore $$x$$ could be $$(2)$$ or $$(-3)$$

I is Not sufficient.

(2) $$x = 4y^4+2y^2+2$$

Value of $$x$$ depends on value of $$y$$. We do not know value of $$y$$, hence we cannot find value of $$x$$ from the equation alone.

II is Not Sufficient.

Combining (1) and (2)

From (1) we get $$x$$ could be $$(2)$$ or $$(-3)$$

From (2) we get $$x$$ should be positive.

Hence $$x$$ should be $$2$$

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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Solve that in a similar way. In condition 2 x must be even since all numbers on the right side are even. So, only possible answer is x = 2 considering condition 1.
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GMAT 1: 790 Q51 V49 GRE 1: Q170 V170 Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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GC2808 wrote:
Solve that in a similar way. In condition 2 x must be even since all numbers on the right side are even. So, only possible answer is x = 2 considering condition 1.

You can only make that assumption if you know the numbers are integers. For instance, here's a much simpler problem:

x = 2y + 4

If you know that x and y are both integers, x definitely has to be even.

But if they don't have to be integers, you could say that y = 0.5. In that case, x = 1 + 4 = 5, which is odd.
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Can someone please tell me if my approach is correct or incorrect:

If we plug in x as -3 into the second statement, we get 4y^4 + 2y^2 + 5 = 0 - but if we try to solve for a value of y, we are unable to because we cannot factor.

If we plug in x as 2 into the second statement, we get 4y^4 + 2y^2 = 0, we get either 2y^2 = 0, which means y must be 0, or we get 2y^2 = -1 which, when we simplify, get a negative root - y = root(-1/2). So only one case will work where y is equal to 0 -- so x must be 2? Is there anything wrong with this approach?

Thank you!
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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Bunuel wrote:
HARRY113 wrote:
Hi,

Can anyone clarify my query here?

Nothing is stated about 'y' rite...??
So y can be irrational, rational, integer, etc.

If i assume an extreme case: y an irrational number, then the answer is E rite?

Please clarify? Should i not take this case into consideration?

For (2): regardless what real number y is (rational, irrational, ...), y in even power would be nonnegative. Thus, $$x = 4y^4+2y^2+2=nonnegative+nonnegative+positive=positive$$.

From (1) x = -3 OR x = 2 and from (2) that x is a positive number, therefore when we combine the statements, x can only be 2. Sufficient.

Hi Bunuel,

If you substitute X=-3 back into the equation, it does not satisfy L.H.S=R.H.S

Can you help me understand, when should we substitute back and when not? I am told that always to substitute back in such equations and in case of MODS.

?
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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GMP12 wrote:
Bunuel wrote:
HARRY113 wrote:
Hi,

Can anyone clarify my query here?

Nothing is stated about 'y' rite...??
So y can be irrational, rational, integer, etc.

If i assume an extreme case: y an irrational number, then the answer is E rite?

Please clarify? Should i not take this case into consideration?

For (2): regardless what real number y is (rational, irrational, ...), y in even power would be nonnegative. Thus, $$x = 4y^4+2y^2+2=nonnegative+nonnegative+positive=positive$$.

From (1) x = -3 OR x = 2 and from (2) that x is a positive number, therefore when we combine the statements, x can only be 2. Sufficient.

Hi Bunuel,

If you substitute X=-3 back into the equation, it does not satisfy L.H.S=R.H.S

Can you help me understand, when should we substitute back and when not? I am told that always to substitute back in such equations and in case of MODS.

?

I assume you are talking about the equation x^2 + x + 10 = 16 . If you substitute x=-3 in this, then LHS = (-3)^2 + (-3) + 10 = 9-3+10 = 16, which is same as RHS

But this same value of x=-3 cannot be substituted in second statement's equation (it wont make sense) x = 4y^4 + 2y^2 + 2 because RHS can never be negative, as other people (including Bunuel) have already explained in this thread.
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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From (1) x = -3 OR x = 2 and from (2) that x is a positive number, therefore when we combine the statements, x can only be 2. Sufficient.

Hi Bunuel,

If you substitute X=-3 back into the equation, it does not satisfy L.H.S=R.H.S

Can you help me understand, when should we substitute back and when not? I am told that always to substitute back in such equations and in case of MODS.

?[/quote]

I assume you are talking about the equation x^2 + x + 10 = 16 . If you substitute x=-3 in this, then LHS = (-3)^2 + (-3) + 10 = 9-3+10 = 16, which is same as RHS

But this same value of x=-3 cannot be substituted in second statement's equation (it wont make sense) x = 4y^4 + 2y^2 + 2 because RHS can never be negative, as other people (including Bunuel) have already explained in this thread.[/quote]

Apologies.. I made a stupid calculation mistake!!
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Re: What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2  [#permalink]

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