Question

alpha+beta=10 and alpha×bta=20 find alpha beta

Answer 1

(alpha+beta)²=100
alpha²+beta ²=100- 2alpha.beta
(alpha-beta)²+2apha.beta=100-40
alpha-beta=√100-40-40
=√20
2√5 ans

ab sahi h!

Answer 2

$given \: = > \\ \alpha + \beta = 10 ......(i) \\ \alpha \beta = 20.......(ii)$


$from (i) \\ \alpha = 10 - \beta .....(iii)$


Putting value of alpha in ( ii )

$\alpha \beta = 20$
$(10 - \beta ) \beta = 20$



$10 \beta - { \beta }^{2} = 20$



${ \beta }^{2} - 10 \beta + 20 = 0$

( a = 1 , b = - 10 , c = 20 )


D = b² - 4ac

D = 100 - 4 × 1 × 20

D = 100 - 80

D = 20



$\beta = \frac{ - b + - \sqrt{d} }{2a}$


= ( 10 ± √ 20 ) / 2

= ( 10 ± 2√5 ) / 2

• When + 2√5 then beta will be :-


$\beta = \frac{10 + 2 \sqrt{5} }{2 } \\ \beta = {5 + \sqrt{5} }$


• When - 2√5 then , beta will be :-

$\beta = \frac{10 - 2 \sqrt{5} }{2} \\ \beta = 5 - \sqrt{5}$



Now , putting the value in ( iii )

$\alpha = 10 - \beta$

$when \: \beta = 5 + \sqrt{5}$


$\alpha = 10 - 5 - \sqrt{5}$

$\alpha = 5 - \sqrt{5}$
• When
$\beta = 5 - \sqrt{5}$

then,

$\alpha = 10 - 5 + \sqrt{5}$

$\alpha = 5 + \sqrt{5}$