If p,q,r,s∈ R, then the equation (x2 + p x + 3 q)(- x2 + r x + q)(- x2 + s x - 2 q)=0 has

Question

If p,q,r,s∈ R, then the equation (x2 + p x + 3 q)(- x2 + r x + q)(- x2 + s x - 2 q)=0 has (A) 6 real roots (B) at least two real roots (C) 2 real and 4 imaginary roots (D) 4 real and 2 imaginary roots

Tardigrade Admin · Accepted Answer

For real roots, p2-12q≥ 0 ...(i) r2+4q≥ 0 ...(ii) s2-8q≥ 0 ...(iii) If q>0 , then (ii) is always true. If q < 0 , then (i) and (iii) are always true. So, the equation has atleast two real roots.

Q. If $p, q, r, s \in R,$ then the equation $(x^{2} + p x + 3 q) (- x^{2} + r x + q) (- x^{2} + s x - 2 q) = 0$ has

$6$ real roots

at least two real roots

$2$ real and $4$ imaginary roots

$4$ real and $2$ imaginary roots

For real roots,
$p^{2} - 12 q \geq 0$ ...(i)
$r^{2} + 4 q \geq 0$ ...(ii)
$s^{2} - 8 q \geq 0$ ...(iii)
If $q > 0$ , then (ii) is always true.
If $q < 0$ , then (i) and (iii) are always true.
So, the equation has atleast two real roots.

Q. If p,q,r,s∈R, then the equation (x2+px+3q)(−x2+rx+q)(−x2+sx−2q)=0 has

6 real roots

at least two real roots

2 real and 4 imaginary roots

4 real and 2 imaginary roots

For real roots, p2−12q≥0 ...(i) r2+4q≥0 ...(ii) s2−8q≥0 ...(iii) If q>0 , then (ii) is always true. If q<0 , then (i) and (iii) are always true. So, the equation has atleast two real roots.

Q. If $p, q, r, s \in R,$ then the equation $(x^{2} + p x + 3 q) (- x^{2} + r x + q) (- x^{2} + s x - 2 q) = 0$ has

$6$ real roots

$2$ real and $4$ imaginary roots

$4$ real and $2$ imaginary roots

For real roots,
$p^{2} - 12 q \geq 0$ ...(i)
$r^{2} + 4 q \geq 0$ ...(ii)
$s^{2} - 8 q \geq 0$ ...(iii)
If $q > 0$ , then (ii) is always true.
If $q < 0$ , then (i) and (iii) are always true.
So, the equation has atleast two real roots.