Question

If $\sqrt{2}$ and $3i$ are two roots of a biquadratic equation with rational coefficients, then its equation is, (where $i^2=-1$)

• A. $x^4-7x^2-18=0$
• B. $x^4-7x^2+18=0$
• C. $x^4+7x^2-18=0$
• D. $x^4+7x^2+18=0$

Answer 1

The correct option is C $x^4+7x^2-18=0$

A quadratic equation with rational coefficients has irrational and imaginary roots in conjugate pairs.
So, if one root is $\sqrt{2}$, then the other root is $-\sqrt{2}$.
The quadratic equation is,
$(x-\sqrt{2})(x+\sqrt{2})=0\Rightarrow x^2-2=0$

If one root of a quadratic equation is $3i$, then the other root is $-3i$.
Sum of roots $=0$,
Product of roots $=3i\times (-3i)=-9i^2=9$
The quadratic equation with the imaginary roots is, $x^2+9=0$
Hence, the required biquadratic equation is, $(x^2-2)(x^2+9)=0$

Hence, the required equation is,
$x^4+7x^2-18=0$