If p,q,r,s are real and `prgt4(q+s)` then show that at least one of the equations `x^2+px+q=0 and x^2+rx+s=0` has real roots.

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.

If p , q , ra n ds are real numbers such that p r=2(q+s), then show that at least one of the equations x^2+p x+q=0 and x^2+r x+s=0 has real roots.

If a , b , c , are real number such that a c!=0, then show that at least one of the equations a x^2+b x+c=0 and -a x^2+b x+c=0 has real roots.

If p , qr are real and p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q=0 are real and unequal.

If p ,q are real p!=q , then show that the roots of the equation (p-q)x^2+5(p+q)x-2(p-q)=0 are real and unequal.

if the difference of the roots of the equation x^(2)-px +q=0 is unity.

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

Show that the roots of the equation x^3 +px^2 +qx +r=0 are in H.P then 2q^3 =9r (pq-3r)

If the equations px^2+2qx+r=0 and px^2+2rx+q=0 have a common root then p+q+4r=

If p ,q , in {1,2,3,4}, then find the number of equations of the form p x^2+q x+1=0 having real roots.