Question

$\alpha ,\beta$ are roots of $y^2-2y-7=0$. Find
a) ${\alpha }^2+{\beta }^2$
b) ${\alpha }^3+{\beta }^3$

• A. (a) $20$ (b) $40$
• B. (a) $18$ (b) $50$
• C. (a) $15$ (b) $35$
• D. (a) $24$ (b) $45$

Answer 1

The correct option is B (a) $18$ (b) $50$

If $\alpha$ and $\beta$ are the roots of $a{y}^2+by+c=0$ , Then

$\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$

For equation $y^2-2y-7=0$

$\alpha +\beta =-\frac{-2}{1}=2$

$\alpha \beta =\frac{-7}{1}=-7$

Now,

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

$2^2={\alpha }^2+{\beta }^2+2(-7)$

${\alpha }^2+{\beta }^2=4+14=18$ .................. (1)

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

$=\left(2\right)(18+7)$

$=2\left(25\right)=50$