Question

If the two equation $x^2-cx+d=0$ and $x^2-ax+b=0$ have one common root and the second has equal roots then $2(b+d)$ is equal to

• A. $0$
• B. $a+c$
• C. $ac$
• D. $-ac$

Answer 1

The correct option is C $ac$

Given equations :

$x^2-cx+d=0$ ....(i)

$x^2-ax+b=0$ ....(ii)

To find : value of $2(b+d)$

solution :

It is given that the two equation and have one common root and the second has equal roots. So, let roots of first equation be $p$ and $q$. Let roots of the second equation be $q$ and $q$.

Now from equation (i), we get

$p+q=c$ ...(iv)

$pq=d$ ...(v)

from equation (ii), we get

$q+q=a$ this implies : $q=a/2$ ....(vi)

$q^2=b$ ....(vii)

So, now using equation (v) and (vii)

$2(b+d)=2(pq+q^2)$

$2(b+d)=2q(p+q)$

Now from eq. (iv) and (vi), we get

$2(b+d)=2ac/2$

$2(b+d)=ac$