Answer:

a²c²x² - ( b^2 - 2ac )( a^2 + c^2 )x + ( b^2 - 2ac )^2 = 0

Step-by-step explanation:

Eq. is ax^2 + bx + c = 0    ⇒ x^2 - (-b/a)x + (c/a) = 0.

Quadratic equation written in the form of x^2 - Sx + P = 0 represent S as sum of roots and P as the product of their roots.

So, here if α and β are roots.

S = α + β = - b / a  

P = αβ = c / a

Therefore,

α^2 + β^2 = (α+β)² - 2αβ

    = ( - b / a )^2 - 2( c / a )

    = b^2 / a^2 - 2c / a

    = ( b^2 - 2ac ) / a^2

1 / α^2 + 1 / β^2

= ( α^2 + β^2 ) / ( αβ )^2

= {(b^2 - 2ac)/a^2}/{(c/a)^2}

= ( b^2 - 2ac ) / c^2

Here, we are asked for the equation with roots ( b^2 - 2ac ) / a^2 and ( b^2 - 2ac ) / c^2.

 So, if that equation is x^2 - Kx + L = 0. So, K is the sum of roots and L is the product of roots.

This means :

K = ( b^2 - 2ac ) / a^2 + ( b^2 - 2ac ) / c^2

K = ( b^2 - 2ac )( a^2 + c^2 ) / ( ac )^2

 And,

L = ( b^2 - 2ac )/a^2 * ( b^2 - 2ac )/c^2

L = ( b^2 - 2ac )^2 / ( ac )^2

HENCE the REQUIRED equation is :

 x^2 - x( b^2 - 2ac )( a^2 + c^2 )/(ac)^2 + ( b^2 - 2ac )^2/(ac)^2 = 0

⇒ a²c²x² - ( b^2 - 2ac )( a^2 + c^2 )x + ( b^2 - 2ac )^2 = 0

Hence the required equation is a²c²x² - ( b^2 - 2ac )( a^2 + c^2 )x + ( b^2 - 2ac )^2 = 0.