Question

if b^3 + a^2c + ac^2 - 3abc = 0 so, prove that one root is square then other root in this equation : ax^2 + bx + c = 0​

Answer 1

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one root of quadratic equation ax² + bx + c =0 is square of other

Let say roots are

m  & m²

m + m² = - b/a  => b = -am(1 + m)

m * m² = c/a   => c = am³

to show

b³ + a²c + ac² =3abc

LHS

= b³ + a²c + ac²

= (-am(1 + m))³  + a²am³  + a(am³)²

= -a³m³(1 + m)³  + a³m³  + a³m⁶

= a³m³(-(1 + m)³ + 1  + m³)

= a³m³(-1 - m³  - 3(1 +m)m + 1  + m³)

= -3a³m³(1 +m)m

= -3a am³ * a(m + m²)

= -3ac (-b)

= 3abc

= RHS