If one root of quadratic equation ax^2 + bx + c =0 is square of other show that b^3 + a^2c + ac^2 =3abc.
Answers
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b³ + a²c + ac² =3abc if one root of quadratic equation ax² + bx + c =0 is square of other
Step-by-step explanation:
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
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