Question

If $\alpha ,\beta$ are the roots of the quadratic equation $4x^2-4x+1=0$, then ${\alpha }^3+{\beta }^3$ is equal to

• A. $1/4$
• B. $1/8$
• C. $16$
• D. $32$

Answer 1

The correct option is A $1/4$

$\Rightarrow$ $\alpha$ and $\beta$ are roots of the equation $4x^2-4x+1=0$.

Here, $a=4,b=-4,c=1$

$\Rightarrow$ $\alpha +\beta =\frac{-b}{a}=\frac{-(-4)}{4}=1$ ---- ( 1 )

$\Rightarrow$ $\alpha \beta =\frac{c}{a}=\frac{1}{4}$ ----- ( 2 )

$\Rightarrow$ $(\alpha +\beta {)}^3={\alpha }^3+{\beta }^3+3{\alpha }^2\beta +3\alpha {\beta }^2$

$\Rightarrow$ $(1{)}^3={\alpha }^3+{\beta }^3+3\alpha \beta (\alpha +\beta )$ [ From ( 1 ) ]

$\Rightarrow$ $1={\alpha }^3+{\beta }^3+3\times \frac{1}{4}\left(1\right)$

$\Rightarrow$ ${\alpha }^3+{\beta }^3=1-\frac{3}{4}$

$\therefore$ ${\alpha }^3+{\beta }^3=\frac{4-3}{4}=\frac{1}{4}$