Question

If x2+5x+7=0 and ax2+bx+c=0 have a common root and a,b,c ∈ N, then the minimum value of a+b+c is .

• A. 1
• B. 5
• C. 7
• D. 13

Answer 1

The correct option is D 13

The roots of equation $x^2+5x+7=0$ are non-real. And we know that if a quadratic equation with real coefficients have non real roots, then they occur in conjugate pair.
We are given that the both equations have a common root and a,b,c are natural numbers. It means the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+c=0$have complex roots in conjugate pair. Thus, the both equations have both common roots.
$\mathrm{So},\frac{a}{1}=\frac{b}{5}=\frac{c}{7}=k,k\in N\Rightarrow a=k,b=5k,c=7k\Rightarrow a+b+c=13k\mathrm{Hence},\mathrm{the}\mathrm{minimum}\mathrm{value}\mathrm{of}a+b+c\mathrm{is}13$