Question

if
$\alpha and \beta$
are the real roots of the equation
${3x}^{2} + 7x - 2 = 0$
find the value of
$\frac{ { \alpha }^{2} }{ \beta } + \frac{ { \beta }^{2} }{ \alpha }$

​

Answer 1

x = 2 as,

3x^2 = 3 * 2^2

= 3 * 4= 12 and 7x = 7 * 2 = 14

= 14 - 2

= 12

therefore 12 -12 = 0

Answer 2

Answer:

$\tt{ \frac{\alpha^{2}}{\beta} + \frac{\beta^{2}}{\alpha}}$

=> $\tt{\frac{\alpha^{3} + \beta^{3}}{\alpha\beta}}$

Identity to be used:

$\tt{(a+b)^{3}=a^{3}+b^{3}+3ab(a+b)}$

=> $\tt{\frac{(\alpha + \beta)^{3} - 3\alpha\beta(\alpha + \beta)}{\alpha\beta}}$

In a polynomial, we know that:

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

$\tt{\alpha\beta = \frac{c}{a}}$

After putting the respective values, we get:

=> $\tt{((\frac{-7}{3})^{3} - 3(\frac{-2}{3})(\frac{-7}{3})) \div (\frac{-2}{3})}$

=> $\tt{\frac{469}{18}}$

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