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05 Dec 2010, 14:28
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
59% (01:14) correct
41%
(01:27)
wrong
based on 302
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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
(1) a = 1
(2) b = 2
05 Dec 2010, 14:51
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ajit257
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.
General Discussion
06 Dec 2010, 22:47
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ajit257
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.
