Question

If $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root. And $a,b\ne 0,$ then find the value of $\frac{a^3+b^3+c^3}{abc}$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C

3

Given,
Two quadratic equations$a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$
having one root in common.

Let common root be $\alpha$.
$\Rightarrow a{\alpha }^2+b\alpha +c=0$ $.......\left(1\right)$
And, $b{\alpha }^2+c\alpha +a=0$ $.......\left(2\right)$

On applying the condition for one common root we get
$(bc-a^2{)}^2=(ca-b^2\left)\right(ab-c^2)$
On simplifying we get
$\Rightarrow (bc{)}^2+(a^2{)}^2-2a^{2}bc=a^{2}bc-c^{3}a-b^{3}a+b^2{c}^2$
$\Rightarrow a^4+a(b^3+c^3)=3a^{2}bc$
$\Rightarrow a[a^3+b^3+c^3]=3a^{2}bc$
$\Rightarrow a^3+b^3+c^3=3abc$
$\Rightarrow \frac{a^3+b^3+c^3}{abc}=3$
Hence, the value of given expression is 3.