What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2

Question

What is the value of x?

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

BrentGMATPrepNow · Accepted Answer

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( 0 ⁴) + 2( 0 ²) + 2. Evaluate to see that  x = 2 
Case b: y =  1 , in which case x = 4( 1 ⁴) + 2( 1 ²) + 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

Answer: C

Cheers,
Brent

Bunuel · Answer

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.

Answer: C.

GMAT4937 · Answer

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.

HARRY113 · Answer

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?

Dirky · Answer

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.

BrentGMATPrepNow · Answer

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

Answer: C

Cheers,
Brent

sashiim20 · Answer

(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

Answer (C)...

GC2808 · Answer

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.

ccooley · Answer

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.

infinitemac · Answer

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!

GMP12 · Answer

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.

?

amanvermagmat · Answer

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.

GMP12 · Answer

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.

Answer: C.

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.

Apologies.. I made a stupid calculation mistake!!

ocelot22 · Answer

(1) (x+3)(x-2) = 0, so x =-3 or x =2 NS

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

ok, y =? NS

(1) and (2). At first glance it may seem there is not enough information here, however:

notice our variables are to even exponents on the RHS of statement 2. that means that the minimum value x can be is 2. sufficient

OA is C

Basshead · Answer

(1) Solving the equation, we get x = -3, 2. INSUFFICIENT.

(2) Clearly insufficient -- we can get different values for y. INSUFFICIENT.

(1&2) Statement 1 tells us that x = -3 or 2. At first glance, the two statements combined might not seem sufficient. However, notice that y is raised to an even exponent, meaning that y is non negative.

Statement 2 actually tells us that x MUST be positive. We only have 1 positive solution, 2. SUFFICIENT.

Answer is C.

ocelot22 · Answer

(1) Subtract 16 and factor : (x+3)(x-2) = 0 NS

(2) let a = y^2: 4a^2 +2a +2=0. Will have nonreal solutions NS

(1) and (2) We know (x+3)(x-2) = 0, so (4a^2+2a+2+3)=0----> non real solution or (4a^2+2a+2-2)=0 ---> 4a^2 +2a=0--> 2a(2a^2+1)=0, so 2a = 0 or (2a^2+1)=0 but nonreal sol, so 2a = 0 so a=0 so y^2=0 s0 x=0+0+2=2 sufficient

OA is E

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What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2

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What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2

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