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If x^2 + 3x + c = (x + a)(x + b) for all x, what is value

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by mitzwillrockgmat » Mon May 24, 2010 11:11 am
If x^2 + 3x + c = (x + a)(x + b) for all x, what is the value of c ?

(1) a = 1
(2) b = 2

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hello! Can someone explain why the answer is D?
I picked C.

x^2 + 3x + c = (x + a)(x + b)
x^2 + 3x + c = (x + 1)(x + 2)
x^2 + 3x + c = x^2 + 3x + 2
So, c = 2

How can both of these stat individually answer??

stat 1) x^2 + 3x + c = (x + 1)(x + b)
x^2 + 3x + c = x^2 + xb + x + b
2x + c = xb + b

to many variables left! the same is with stat 2! HELP PLEASE! :)
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by Patrick_GMATFix » Mon May 24, 2010 11:24 am
This tests whether you can factor quadratics. For instance, x^2 + 3x - 10 = (x-2)(x+5) because the two numbers (-2) and (5) add up to the number in front of x, and multiply to the constant.

the way factoring works, to factor x^2 + 3x + c, we must write (x+a)(x+b) where a and b are two numbers such that
  • a + b = 3
  • ab=c
So we already have one known equation (a+b=3). If we're given the value of one of them we can find the other easily.

The question asks for c. REPHRASE: What is ab?

(1) By plugging a = 1 into the first equation (a+b=3), we can find b and solve for c = ab.
(2) same as 1.

D is correct.

If you have access to the Solutions Engine, run a practice drill with topic='Algebraic Translations' and difficulty='500-600 AND 600-700' to see similar questions.

Best of luck,
-Patrick
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by tpr-becky » Mon May 24, 2010 1:09 pm
This question tests the basics of FOIL on quadratics - according to FOIL rules you know that ab=c and that a+b=3

So we are looking for the value of ab and we know that a+b=3

statement 1 says a=1 therefore we know that b=2 and thus can get the value of ab.

AD

statement 2 says b=2 so a+2=3 which means that a=1 same as above

therefore the answer is D.
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by indiantiger » Mon May 24, 2010 3:49 pm
If x^2 + 3x + c = (x + a)(x + b) for all x, what is the value of c ?

(1) a = 1
(2) b = 2

First lets try to make the LHS and RHS same in terms of power i.e. make both sides quadratic
LHS = x^2 + 3x + c
RHS we need to solve
= x^2+(a+b)x+ab
LHS = RHS
which gives us
a+b = 3-------(A)
and
c = ab

statement 1) a = 1
put the value of a =1 in equation (A)
we get b = 2
which gives c = 2

sufficient

Statement 2) b= 2
put the value of b =2 in equation (A)
we get a = 1
which gives c = 2

sufficient

Hence (D)
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