In the expression #12x^2+ ax -20#, a is an integer. If 3x+4 is a factor of the expression above, what is the value of a?

2 Answers
Narad T.
Oct 19, 2016

a=1

Explanation:

let #f(x)=12x^2+ax-20#
then if #(3x+4)# is a factor
#f(-4/3)=12*16/9-4/3a-20=0#
Simplifying
#64/3-4/3a-20=0#
So #4/3a=64/3-20=4/3#
#a=1#
So #12x^2+x-20=(3x+4)(4x-5)#

Cesareo R.
Oct 19, 2016

#a=1#

Explanation:

If #3x+4# is a factor then

#12x^2+a x-20=(3x+4)(bx+c)#

Now choosing #a,b,c# such that this relationship is verified for all #x in RR# implies in the conditions:

#{(20 + 4 c=0), (a - 4 b - 3 c=0), (12 - 3 b=0):}#

solving for #a,b,c# we obtain

#a=1,b=4,c=-5#

so #a=1#

NOTE:

#(3 x + 4) (b x + c) = 3bx^2+(4b+3c)x+4c# and then

#12x^2+a x-20= 3bx^2+(4b+3c)x+4c# or

#(12-3b)x^2+(a-(4b+3c))x - (20+4c)=0#

This equality must be verified for all #x in RR# then as a consequence

#{(20 + 4 c=0), (a - 4 b - 3 c=0), (12 - 3 b=0):}#

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