Question

If $x+a$ is a common factor of $f\left(x\right)=x^2+x-6$ and $g\left(x\right)=x^2+3x-18$ then find the value of $a$.

• A. $1$
• B. $2$
• C. $3$
• D. No value of a exist

Answer 1

Answer: (D)

No value of a exist

Given that:

$f\left(x\right)=x^2+x-6$ and $g\left(x\right)=x^2+3x-18$

Since $x+a$ is common factor of $f\left(x\right)\&g\left(x\right)$ than $f(-a)\&g(-a)$ will be zero.

So,

$f(-a)=(-a{)}^2+(-a)-6=0$$\Rightarrow f(-a)=a^2-a-6=0$

So, $a^2-a-6=0\Rightarrow a^2-3a+2a-6=0$

$\Rightarrow a(a-3)+2(a-3)=0$

$\Rightarrow (a+2)(a-3)=0$

So $a=3,-\mathrm{2........}\left(1\right)$

Since

$g(-a)=(-a{)}^2+3(-a)-18=0\Rightarrow a^2-3a-18=0$

$\Rightarrow a^2-6a+3a-18=0$

$a(a-6)+3(a-6)=0$

$(a-6)(a+3)=0$

$a=6,-\mathrm{3.........}\left(2\right)$

compering equation 1 and 2.

$a$ has no solution.