If p,q,r,s∈R, then the equation x^2+px+3q-x^2+rx+q-x^2+sx-2q=0 has

Correct answer: (B) at least two real roots. For real roots, p^2-12q≥0 ...(i) r^2+4q≥0 ...(ii) s^2-8q≥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.

Q3 / 82 · NTA Abhyas (JEE Main) 2020 · Mathematics · Quadratic Equation

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

A
$6$ real roots
Correct answer
B
at least two real roots
C
$2$ real and $4$ imaginary roots
D
$4$ real and $2$ imaginary roots

Solution

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.

Hence, the correct answer is option: 'B'.

If p,q,r,s∈R, then the equation x2+px+3q-x2+rx+q-x2+sx-2q=0 has

Solution

More Quadratic Equation Questions

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