Question

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?

Answer 1

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)`

Answer 2

`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):}`