Question

if alpha and beta are the zeros of the quadratic polynomial 3 X ^2 + 5 x + 7 evaluate 1. 1 by alpha + 1 by beta 2. 1 alpha ^2 + 1 betta ^2​

Answer 1

Answer :

1. 1/α + 1/β = -5/7
2. 1/α² + 1/β² = -17/49

Solution :

3x² + 5x + 7 = 0

α + β = -b/a = -5/3

αβ = c/a = 7/3

↠ $\frac{1}{ \alpha } + \frac{1}{ \beta }$

↦$\frac{ \alpha + \beta }{ \alpha \beta }$

↦$\frac{ \frac{ - 5}{3} }{ \frac{7}{3} }$

↦$\frac{ - 5}{7}$

↠$\frac{1}{ { \alpha }^{2} } + \frac{1}{ { \beta }^{2} }$

↦$\frac{ { \alpha }^{2} + { \beta }^{2} }{ ( \alpha \beta)^{2} }$

↦$\frac{( \alpha + \beta )^{2} - 2 \alpha \beta }{( \alpha \beta)^{2} }$

↦$\frac{( { \frac{ - 5}{3}) }^{2} - 2 (\frac{7}{3}) }{( { \frac{7}{3} )}^{2} }$

↦$\frac{(25 - 42)}{49}$

↦$\frac{ - 17}{49}$