Question

If $\alpha ,\beta$ are the roots of the equation $2x^2+3x+4=0,$ find $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

• A. $\frac{7}{4}$
• B. $\frac{-7}{8}$
• C. $\frac{7}{8}$
• D. $\frac{-7}{4}$

Answer 1

The correct option is B

$\frac{-7}{8}$

$2x^2+3x+4=0,$
$\alpha +\beta =\frac{-3}{2}\alpha \beta =\frac{4}{2}=2\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{{(\alpha +\beta )}^2-2\alpha \beta }{\alpha \beta }=\frac{{\left(\frac{-3}{2}\right)}^2-2\times 2}{2}=\frac{(\frac{9}{4}-4)}{2}=\frac{-7}{8}$

Note that even though the numerator was some of squares and the denominator was positive, we got a negative value. This is because the roots of the equation are complex numbers.