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 the other two roots are

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

Answer 1

The correct option is C

Complex Number

$x^4$ - 2 $x^3$ -2 $x^2+18x-63=0$ _____ (1)

Let's roots are $\alpha ,\beta ,\gamma ,\delta$

Given $\alpha +\beta =0$

Sum of all the roots $\alpha +\beta +\gamma +\delta =2,\gamma +\delta =2$

Let's assume product of first two roots = p

$\alpha \beta =pand\gamma \delta =q$

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.