Home
Class 12
MATHS
Show that if p,q,r and s are real number...

Show that if `p,q,r` and `s` are real numbers and `pr=2(q+s)`, then atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.

Text Solution

Verified by Experts

Let `D_(1)` and `D_(2)` be the discriminants of the give equations `x^(2)+px+q=0` and `x^(2)+rx+s=0`, respectively.
Now `D_(1)+D_(2)=p^(2)-4q+r^(2)-4s=p^(2)+r^(2)-4(q+s)`
`=p^(2)+r^(2)-2pr` [given `pr=2(q+s)`]
`=(p-r)^(2)ge0` [ `:'p` and `q` are real]
or `D_(1)+D_(2)ge0`
Hence, atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.
Promotional Banner

Similar Questions

Explore conceptually related problems

If p, q, r are real numbers then:

Real roots of the equation x^(2)+6|x|+5=0 are

If p and q are the roots of the equation x^(2)+p x+q=0 then

The sum of the real roots of the equation x^(2)+|x|-12=0 is

If p, q, are real and p ne q then the roots of the equation (p-q) x^(2)+5(p+q) x-2(p-q)=0 are

If p and q are the roots of the equation x^2-p x+q=0 , then

The roots of the equation (q- r) x^(2) + (r - p) x + (p - q)= 0 are

If one root of the equation x^(2) + px + q = 0 is square of the other root, then :

If a, b, c are positive real numbers, then the number of real roots of the equation a x^(2)+b|x|+c=0 is

The number of real roots of the equation |x|^(2)-3|x|+2=0 is