[" If "a_(1)x^(2)+b_(1)x+c_(1)=0" and "a_(2)x^(2)+b_(2)x+c_(2)=0" have a "],[" common root then that common root is "]

If a_(1)x^(2)+b_(1)x+c_(1)=0 and a_(2)x^(2)+b_(2)x+c_(2)=0 has a common root,then the common root is

If two equation a_(1) x^(2) + b_(1) x + c_(1) = 0 and, a_(2) x^(2) + b_(2) x + c_(2) = 0 have a common root, then the value of (a_(1) b_(2) - a_(2) b_(1)) (b_(1) c_(2) - b_(2) c_(1)) , is

If two equation a_(1) x^(2) + b_(1) x + c_(1) = 0 and, a_(2) x^(2) + b_(2) x + c_(2) = 0 have a common root, then the value of (a_(1) b_(2) - a_(2) b_(1)) (b_(1) c_(2) - b_(2) c_(1)) , is

If a_1x^2 + b_1 x + c_1 = 0 and a_2x^2 + b_2 x + c_2 = 0 has a common root, then the common root is

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

The equations x^(2)-2x+1=0,x^(2)-3x+2=0 have a common root then that common root is

Suppose two quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0 and a_(2)x^(2)+b_(2)x+c_(2)=0 have a common root alpha , then a_(1)alpha^(2)+b_(1)alpha+c_(1)=0 " " …..(1) and a_(2)alpha^(2)+b_(2)alpha+c_(2)=0 " " ……(2) Eliminating alpha using crose - multiplication method gives us the condition for a common root Solving two equations simultaneously, the common root can be obtained. Now, consider three quadratic equations, x^(2)-2r p_(r )x+r=0, r=1, 2, 3 given that each pair has exactly one root common. In the notations of above problem, (gamma)/(alpha) is equal to

Theorem : A necessary and sufficient condition for the quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0" and "a_(2)x^(2)+b_(2)x+c_(2)=0 to have a common roots is (c_(1)a_(2)-c_(2)a_(1))^(2)=(a_(1)b_(2)-a_(2)b_(1))(b_(1)c_(2)-b_(2)c_(1))