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Tricky Algebra Data sufficiency problem

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Tricky Algebra Data sufficiency problem

by psriniva » Tue Apr 29, 2008 2:20 pm
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(A) d = 3
(B) b = 6

Resolving the equation produces
bx +c = 2dx + d^2 for all values of x.

Analysis 1.
If we now conclude that 2d = b and c = d^2.
So A is sufficient because c = d^2 = 9.
B is sufficient because 2d = 6, so d = 3, so c = 9.
Thus the answer is D - both sufficient individually.

Analysis 2.
If we can not conclude that 2d = b and c = d^2, then
further resolution of the equation for c produces the following.
c = (2d-b)x + d^2.
So, for x = 0, c = d^2. Hence c = 9. What is true for one value of x must be true for all values of x according to question stem. So, A is sufficient.
For all other values of x, such as x = 1, 2 etc. both b and d need to be known. So, B by itself is not sufficient.
So, the answer is A.

Which answer is correct, A or D?
Paddy Srinivas
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Re: Tricky Algebra Data sufficiency problem

by Sunny22uk » Tue Apr 29, 2008 3:09 pm
Answer should be A, Analysis1 for Option B is wrong :-"B is sufficient because 2d = 6, so d = 3, so c = 9" The 2d=6 option is not given anywhere for the question if you are just looking at option B
psriniva wrote:If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(A) d = 3
(B) b = 6

Resolving the equation produces
bx +c = 2dx + d^2 for all values of x.

Analysis 1.
If we now conclude that 2d = b and c = d^2.
So A is sufficient because c = d^2 = 9.
B is sufficient because 2d = 6, so d = 3, so c = 9.
Thus the answer is D - both sufficient individually.

Analysis 2.
If we can not conclude that 2d = b and c = d^2, then
further resolution of the equation for c produces the following.
c = (2d-b)x + d^2.
So, for x = 0, c = d^2. Hence c = 9. What is true for one value of x must be true for all values of x according to question stem. So, A is sufficient.
For all other values of x, such as x = 1, 2 etc. both b and d need to be known. So, B by itself is not sufficient.
So, the answer is A.

Which answer is correct, A or D?
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Re: Tricky Algebra Data sufficiency problem

by Sunny22uk » Tue Apr 29, 2008 3:09 pm
Answer should be A, Analysis1 for Option B is wrong :-"B is sufficient because 2d = 6, so d = 3, so c = 9" The 2d=6 option is not given anywhere for the question if you are just looking at option B
psriniva wrote:If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(A) d = 3
(B) b = 6

Resolving the equation produces
bx +c = 2dx + d^2 for all values of x.

Analysis 1.
If we now conclude that 2d = b and c = d^2.
So A is sufficient because c = d^2 = 9.
B is sufficient because 2d = 6, so d = 3, so c = 9.
Thus the answer is D - both sufficient individually.

Analysis 2.
If we can not conclude that 2d = b and c = d^2, then
further resolution of the equation for c produces the following.
c = (2d-b)x + d^2.
So, for x = 0, c = d^2. Hence c = 9. What is true for one value of x must be true for all values of x according to question stem. So, A is sufficient.
For all other values of x, such as x = 1, 2 etc. both b and d need to be known. So, B by itself is not sufficient.
So, the answer is A.

Which answer is correct, A or D?
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Tricky Algebra Data sufficiency problem

by psriniva » Tue Apr 29, 2008 6:06 pm
I am posting the question and my elaborate response to Sunny's reply below. Please read my revised analysis one more time and let me know your comments.

Here is the question.
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(A) d = 3
(B) b = 6

Resolving the equation in the question stem produces
bx +c = 2dx + d^2 for all values of x.


Analysis 1.
From the reduced equation of the question stem alone,
bx + c = 2dx + d^2 and without looking at either A or B,
if we now conclude that
a. 2d = b and
b. c = d^2.
I am concluding this because Coefficient of x should be same on both sides of the equation if the equation holds true for all values of x.
So A is sufficient because c = d^2 = 9.
B is sufficient because from the question stem alone we know that b = 2d. From B we know that b = 6. So, using the question stem and B, we know that 2d = 6, so d = 3, so c = 9. Thus B is sufficient as well.
Thus the answer is D - both sufficient individually.

Analysis 2.
If we can not conclude that 2d = b and c = d^2, from the reduced equation of the question stem, then
further resolution of the equation for c produces the following.
c = (2d-b)x + d^2.
So, for x = 0, c = d^2. Hence c = 9. What is true for one value of x must be true for all values of x according to question stem. So, A is sufficient.
For all other values of x, such as x = 1, 2 etc. both b and d need to be known. So, B by itself is not sufficient.
So, the answer is A.

Which answer is correct, A or D?
Paddy Srinivas
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by netigen » Tue Apr 29, 2008 7:28 pm
Answer is D

solve the second equation to get x^2+2dx+d^2

equating the result above to the first equation gives us

b = 2d
c = d^2

Note if you equate c = d^2 as in your analysis (2) then b has to be 2d for all values of x because for any other value of b it will be a different curves on the XY plane.

from A we know that d=3 so c = 9
from B we know that b=6=2d hence d = 3 and c = 9

both are sufficient to ans the question.
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Elaboration on analysis 2

by rs007 » Wed Apr 30, 2008 3:37 pm
psriniva,
I agree with part of your analysis 2

----------------------------------------------------------------------------------------
Analysis 2.
If we can not conclude that 2d = b and c = d^2, from the reduced equation of the question stem, then
further resolution of the equation for c produces the following.
c = (2d-b)x + d^2.
So, for x = 0, c = d^2. Hence c = 9. What is true for one value of x must be true for all values of x according to question stem.
----------------------------------------------------------------------------------------

Now, since as you mentioned:
c = (2d-b)x + d^2 must hold of all values of x.
using x=0, we get c = d^2.
However, since c is a constant, it can have only one value (d^2).
For this to occur across all values of x, 2d-b must be equal to 0. This leads to 2d=b.
but we know c=d^2 so c = (b/2)^2

So answer has to be D.

What do you think?
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