Question

If the equations $x^2+4x+5=0$ and $a{x}^2+bx+c=0$ have a common root (where $a,b,c\in N$), then the least value of $a+b+c$ is equal to:

• A. $10$
• B. $15$
• C. $20$
• D. $25$

Answer 1

The correct option is A $10$

The root of the given equation $x^2+4x+5=0$ are

$x=\frac{-4\pm \sqrt{16-20}}{2}=-2\pm i$, where $i=\sqrt{-1}$.

Now, for the equation $a{x}^2+bx+c=0$, where $a,b,c$ are positive integers, has a common root with the first equation.

As the two roots of the first equation are complex and $a,b,c$ are natural numbers, the second equation also will have the same roots.

As complex roots occurs in pairs of a quadratic equation with real co-efficient.

For least value of $a,b,c$ we will get least $a+b+c$

For that case the second equation will be same as the first

So, $a+b+c=1+4+5=10$.