If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?

Question

GMATH  practice exercise (Quant Class 14)

If  \,p\left( x \right) = k{x^2} - kx - 12\,  for all values of  x , what is the value of the constant  k  ?

(1)  p\left( 3 \right) = 0

(2)  p(x)  is divisible by  x+2

chetan2u · Answer

At the very first look, you should be tempted towards D.

(1)  p\left( 3 \right) = 0 
  Putting x as 3, the quadratic equation would convert into a linear equation with just one variable and so should be sufficient  
Put x=3 so  p\left( 3 \right) = 0.......\,p\left( 3 \right) =0= k{3^2} - k3 - 12\.......9k-3k=12...k=2 
Sufficient

(2)  p(x)  is divisible by  x+2 
  This should tell you to look if the equation can be converted in terms of x+2, and divisible by x+2 means substituting x as -2, we should get the equation as 0   
 k{x^2} - kx - 12\,=k(-2)^2-(-2)k-12=0........4k+2k-12=0...k=2 
Sufficient

D

KanishkM · Answer

From 1 
when we substitute the value as 3 in the expression  \,p\left( x \right) = k{x^2} - kx - 12\,  
We get a definite value for k

From 2 
x + 2 means that x =-2, will satisfy the expression, which can be equated to 0
Again we will get a definite value for k

D

fskilnik · Answer

Obs.: although the above version of the question stem is a bit "wordy", we believe it´s better (mathematically speaking).

p\left( x \right) = k{x^2} - kx - 12\,\,\,\,\,\left( * \right)

? = k

\left( 1 \right)\,\,\,0\,\, = \,\,p\left( 3 \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( 3 \right)^2} - k\left( 3 \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.

\left( 2 \right)\,\,\,0 = p\left( { - 2} \right)\,\,\mathop = \limits^{\left( * \right)} \,\,k{\left( { - 2} \right)^2} - k\left( { - 2} \right) - 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,k\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}

The correct answer is therefore (D).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

bumpbot · Answer

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If p(x) = kx^2 - kx - 12, for all values of x, what is the value of k?

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