Question

If $x+k$ is the HCF of $x^2+ax+b$ and $x^2+cx+d$ then what is the value of K

• A. $\frac{d-b}{c-a}$
• B. $\frac{a-b}{c-d}$
• C. $\frac{a+b}{c-d}$
• D. none of these

Answer 1

Answer: (A)

$\frac{d-b}{c-a}$

Given $x+k$ is HCF of $x^2+ax+b$ and $x^2+cx+d$

Therefore $x+k$ is a factor of each expression.

Put $x=-k$ is the given expressions.

$(-k{)}^2+(a\left)\right(-k)+b=0$

$\Rightarrow k^2-ka+b=0$ ………$\left(1\right)$

also $(-k{)}^2+c(-k)+d=0$

$\Rightarrow k^2-ck+d=0$ …..$\left(2\right)$

substract equation $\left(2\right)$ from $\left(1\right)$

$c.k-a.k-d+b=0$

$k(c-a)=d-b$

$\Rightarrow k=\frac{d-b}{c-a}$.