If `a,b,c, a_1,b_1,c_1` are rational and equations `ax^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0` have one and only one root in common, prove that `b^2-ac and b_1^2-a_1c_1` must be perfect squares.

If a,b,c in R and the equation ax^(2)+bx+c=0 and x^(2)+x+1=0 have a common root,then

If the equations ax^(2)+bx+c=0 and x^(2)+x+1=0 has one common root then a:b:c is equal to

If x^(2)+ax+bc=0 and x^(2)+bx+ca=0 have a common root,then a+b+c=1

If the equations ax^(2) +2bx +c = 0 and Ax^(2) +2Bx+C=0 have a common root and a,b,c are in G.P prove that a/A , b/B, c/C are in H.P

If ax^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0 have commonroot and a/a_1,b/b_1,c/c_1 are in A.P., show that are: ax^2+2bx+c=0

If the quadratic equations x^(2)+bx+c=0 and bx^(2)+cx+1=0 have a common root then prove that either b+c+1=0 or b^(2)+c^(2)+1=bc+b+c

If ax^2+bx+c=0 and px^2+qx+r=0 have one and only one root in common and a,b,c,p,q,r being rational, then b^2-ac and q^2-pr are 1) both are perfect squares 2) b^2-ac is a perfect square but q^2-pr is not a perfect square 3) q^2-pr is a perfect square but b^2-ac is not a perfect square 4) both are not perfect squares

If b_1b_2=2(c_1+c_2) , then at least one the of the equations x^(2) +b_1x+c_1=0 and x^(2)+b_2x+c_2=0 has :

If (a_(1))/(a_(2)),(b_(1))/(b_(2)),(c_(1))/(c_(2)) are in AP and the equations a_(1)x^(2)+2b_(1)x+c_(1)=0 and a_(2)x^(2)+2b_(2)x+c_(2)=0 have a common root, prove that a_(2),b_(2),c_(2) are in G.P