Question

For equation $x^4-2x^3-2x^2+18x-63=0$ if two of its roots are equal in magnitude but opposite in sign. Then other two roots are

• A. Imaginary
• B. Real and negative
• C. None of these
• D. Real and positive

Answer 1

The correct option is A

Imaginary

Given: $x^4-2x^3-2x^2+18x-63=0...\left(1\right)$

Let roots be $\alpha ,\beta ,\gamma ,\delta .$

It is given that $\alpha +\beta =0$

Using the relation between roots and coefficeints of a polynimial equation
$\alpha +\beta +\gamma +\delta =2\Rightarrow \gamma +\delta =2$

Let $\alpha \beta =p$ and $\gamma \delta =q$

Bi-quadratic equation ( $x^2$ + p) ( $x^2-2x+q)$ ______ (2)

Compare the coefficient of equation 1 and 2

p + q = - 2, -2p = 18, pq = - 63

p = -9, q = 7

pq = - 63 (satisfies)

Biquadratic equation is

( $x^2$ - 9) ( $x^2-2x+7)=0$

Roots are $\pm 3and1\pm i\sqrt{6}$ Other two roots are complex number.