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

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ajit257
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If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink] New post   05 Dec 2010, 14:28
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If x^2 + 3x + c = (x + a)(x + b) for all x, what is the value of c ?

(1) a = 1
(2) b = 2
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If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink] New post   05 Dec 2010, 14:51
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ajit257 wrote:
If x2 + 3x + c = (x + a)(x + b) for all x, what is the value of c ?
(1) a = 1
(2) b = 2


Since \(x^2+3x+c=(x + a)(x + b)\) is true "for ALL values of \(x\)" then it must hold true for \(x=0\) too --> \(c=ab\);

Next, substitute \(c=ab\) --> \(x^2+3x+ab=x^2+bx+ax+ab\) --> \(3x=bx+ax\) --> again it must be true for \(x=1\) too --> \(a+b=3\).

So we have: \(c=ab\) and \(a+b=3\). Question: \(c=?\)

(1) \(a=1\) --> as \(a+b=3\) then \(b=2\) and \(c=ab=2\). Sufficient
(2) \(b=2\) --> as \(a+b=3\) then \(a=1\) and \(c=ab=2\). Sufficient.

Answer: D.

Similar questions:
equation-ds-105823.html?hilit=constants#p768377
equating-coefficients-in-a-quadratic-equation-103919.html

Hope it's clear.
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Re: Algebra [#permalink] New post   06 Dec 2010, 22:47
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ajit257 wrote:
If x2 + 3x + c = (x + a)(x + b) for all x, what is the value of c ?
(1) a = 1
(2) b = 2


Any expression of the form \((x + a)(x + b)\) can be expressed as \(x^2 + (a+b)x + ab\).
Comparing this with the quadratic equation containing c, \(a+b = 3\) and \(ab = c\).
So if the value of either \(a\) or \(b\) is known the value of other can be found out from \(a+b = 3\), and putting the two values in \(ab = c\), value of \(c\) can be found out.

Answer D.
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Re: If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink] New post   13 Sep 2016, 07:01
1. a=1. since the coefficient for 3x is known - 3, it must be true that b is 2. we can find a definite answer.
2. b=2. same process is applied. only one value for c is possible.
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Re: If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink] New post   05 Feb 2020, 02:31
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\(x^2\) + 3x + c = (x + a)(x + b) = \(x^2+(a+b)x + ab\)
Comparing
a+b = 3 and ab =c

(1) When a=1, b = 2, c = 2
Sufficient

(2) When b = 2, a=1, c=2
Sufficient

D is correct
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Re: If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink] New post   31 May 2022, 02:36
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Re: If x^2 + 3x + c = (x + a)(x + b) for all x, what is the [#permalink]
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