Question

If one root of $a{x}^2+bx+c=0$ be square of the other, then the value of $b^3+a{c}^2+a^{2}c$, is

• A. $3abc$
• B. $-3abc$
• C. $0$
• D. None of these

Answer 1

The correct option is A $3abc$

$a{x}^2+bx+c=0$

Let, $\alpha ,{\alpha }^2$ be the roots.

$\therefore \alpha +{\alpha }^2=\frac{-b}{a}$ (equation $01$)

$\alpha \cdot {\alpha }^2=\frac{c}{a}$ (equation $02$)

Taking cube of equation $01$

${\alpha }^3+{\alpha }^6+3{\alpha }^3(\alpha +{\alpha }^2)=\frac{-b^3}{a^3}$

$\frac{c}{a}+\frac{c^2}{a^2}+\frac{3c}{a}\left(\frac{-b}{a}\right)=\frac{-b^3}{a^3}$ (equation $03$)

Multiply by $a^3$

$a^{2}c+a{c}^2-3abc=b^3$

$⟹b^3+a^{2}c+a{c}^2=3abc$