Question

If $a,b,c$ are non-zero, unequal rational numbers, then the roots of the equation $ab{c}^2{x}^2+(3a^2+b^2)cx-6a^2-ab+2b^2=0$ are

• A. rational
• B. imaginary
• C. irrational
• D. none of these

Answer 1

The correct option is A rational

Discriminant $\Delta$ of a quadratic equation is $\Delta =B^2-4AC$
$=c^2{(3a^2+b^2)}^2+4ab{c}^2(6a^2+ab-2b^2)=c^2(9a^4+b^4+6a^2{b}^2+24a^{3}b+4a^2{b}^2-8a{b}^3)=c^2(9a^4+b^4+16a^2{b}^2+24a^{3}b-6a^2{b}^2-8a{b}^3)=c^2{\left(3a^2\right)}^2+{(-b^2)}^2+{\left(4ab\right)}^2+2\left(3a^2\right)(-b^2)+2{(-b^2)}^2\left(4ab\right)+2\left(3a^2\right)\left(4ab\right)={(3a^{2}c-b^{2}c+4ab{c}^2)}^2$
Above equation is a perfect square.
Hence, roots are rational.