Question

3s2-6s+4 find the value of alpha / beta + beta / alpha +2 ( 1/alpha + 1/ beta)+ 3alpha beta

Answer 1

Answer:

$\frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } + 2\bigg( \frac{1}{ \alpha } + \frac{1}{ \beta } \bigg) + 3 \alpha \beta = 8$

Given that

$3 {s}^{2} - 6s + 4$
equation has two zeros,namely
$\alpha \: \: and \: \: \beta$
from coefficient and zeros relation

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

$a = 3 \\ \\ b = - 6 \\ \\ c = 4$
$\alpha + \beta = \frac{6}{3} = 2 \\ \\ \alpha \beta = \frac{4}{3}$
To find

$\frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } + 2\bigg( \frac{1}{ \alpha } + \frac{1}{ \beta } \bigg) + 3 \alpha \beta \\ \\ \\ = > \frac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta } + 2\bigg( \frac{ \alpha + \beta }{ \alpha \beta } \bigg) + 3 \alpha \beta$
we don't have the value of
${ \alpha }^{2} + { \beta }^{2}$
to find that

$\alpha + \beta = 2 \\ \\ square \: both \: side \\ \\ { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta = 4 \\ \\ { \alpha }^{2} + { \beta }^{2} = 4 - 2 \alpha \beta \\ \\ { \alpha }^{2} + { \beta }^{2} = 4 - 2 \times \frac{4}{3} \\ \\ { \alpha }^{2} + { \beta }^{2} = 4 - \frac{8}{3} \\ \\ { \alpha }^{2} + { \beta }^{2} = \frac{ 4 }{3}$
Now put all these values

$\frac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta } + 2\bigg( \frac{ \alpha + \beta }{ \alpha \beta } \bigg) + 3 \alpha \beta \\ \\ \\ = > \frac{ \frac{4}{3} }{ \frac{4}{3} } + 2\bigg( \frac{2}{ \frac{4}{3} } \bigg) + 3 \times \frac{4}{3} \\ \\ \\ = > 1 + \frac{4 \times 3}{4} + 4 \\ \\ \\ = > 1 + 3 + 4 \\ \\ = > 8$
Hope it helps you.

Answer 2

Answer:

Given, α and β are the zeroes of the quadratic polynomial p(s) = 3s2 – 6s + 4

Now, α + β = -(-6)/3

=> α + β = 6/3

=> α + β = 2

and α * β = 4/3

Now, (α/β) + (β/α) + 2(1/α + 1/β) + 3 * α * β

= (α2 + β2 )/α * β + 2(α + β)/α * β + 3 * α * β

= {(α + β)2 - 2α * β}/α * β + 2(α + β)/α * β + 3 * α * β

= {22 - 2 *4/3 }/(4/3) + (2 * 2)/(4/3) + 3 * (4/3)

= (4 - 8/3)/(4/3) + 4/(4/3) + 4

= (4/3)/(4/3) + (4*3)/4 + 4

= 1 + 3 + 4

= 8

Step-by-step explanation: