If p, q, r are rational then show that `x^(2)-2px+p^(2)-q^(2)+2qr-r^(2)=0`

If a+b+c=0 , prove that the roots of ax^(2)+bx+c=0 are rational. Hence, show that the roots of (p+q)x^(2)-2px+(p-q)=0 are rational.

If p = 4, q= 3 and r = -2, find the values of : (p^(2)+q^(2)-r^(2))/(pq+qr-pr)

if q is the mean proportional between p and r , prove that : p^(2) - q^(2) + r^(2) = q^(4) ((1)/(p^(2))-(1)/(q^(2)) + (1)/(r^(2))) .

p, q, r and s are integers. If the A.M. of the roots of x^(2) - px + q^(2) = 0 and G.M. of the roots of x^(2) - rx + s^(2) = 0 are equal, then

If two distinct chords, drawn from the point (p,q) on the circle x^(2)+y^(2)=px+qy, where pq!=0 are bisected by x-axis then show that p^(2)gt8q^(2) .

If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots

If p, q, r, s in R , then equaton (x^2 + px + 3q) (-x^2 + rx + q) (-x^2 + sx-2q) = 0 has

If p, q, r are in G.P. and the equations, p x^2+2q x+r=0 and dx^2+2e x+f=0 have a common root, then show that d/p , e/q , f/r are in A.P.

If p/a + q/b + r/c=1 and a/p + b/q + c/r=0 , then the value of p^(2)/a^(2) + q^(2)/b^(2) + r^(2)/c^(2) is:

Factorise : (p^(2) + q^(2) - r^(2))^(2) - 4p^(2)q^(2) .