# Pls answer my question 1)find the quadratic polynomial with zeroes 3+√2 & 3-√2. 2)if alpha*beta are zeroes of x^2 +7x+12,then find the value of (1/2)alpha +(1/2)beta-2(alpha*beta). 3)if alpha ,beta are the ratio of quadratic polynomial p(x)=x^2-(k-6)x+2(2k-1). find the value of k ,if alpha +beta =(1/2alpha*beta).

2
by sanjays2402

2015-04-26T19:33:58+05:30
Let 3+root2 = alpha      and      3 - root 2 = beta
then
alpha + beta = ( 3 + root 2 ) + ( 3 - root 2 )
=  3 + root 2 + 3 - root 2
[ + root 2 and - root 2 gets cancelled ]
= 3 + 3
= 6
alpha * beta = ( 3 + root 2) ( 3 - root 2 )
= 3( 3 - root 2 ) + root 2 ( 3 - root 2 )
= 9 - 3 root 2 + 3 root 2 - 2
[ -3 root 2 and + 3 root 2 gets cancelled ]
= 9 - 2
= 7

rule : -          x square + ( alpha + beta ) x - (alpha*beta)

= x square + 6 x - 7         is the required polynomial
completed
I Mean 2&3
oh those got 2 check it . ill try those
OK!
Pls do fast!
2015-04-27T00:44:34+05:30

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1) sum of roots = 3 + √2 + 3 - √2 = 6
product of roots = (3+√2) (3-√2) = 3² - 2 = 7
quadratic polynomial =  x² - sum of roots + product of roots
x² - 6 x + 7

2)   α + β = - coefficient of x / coefficient of x² = -7
α β = constant term / coefficient of x² = 12
1/2 α + 1/2 β - 2(α β) = 1/2 (α + β) - 2 αβ = -7/2 - 2 * 12 = -55/2

3)  x² - (k-6) x + 2 (2k-1)
α + β = k  - 6
α β = 4 k - 2

as α + β = 1/ (2αβ) ,         k - 6 = 1/(8 k -4)

8 k² - 4 k - 48 k + 24 = 1
8 k² - 52 k  + 23 = 0
factors of  23 * 8 =  23 * 2 * 6 =  46 * 6 whose sum is 52.
8k² - 4 k - 46 k + 23 = 0
4 k (2 k - 1) - 23 (2k - 1) = 0
(4k - 23) ( 2k - 1) = 0

k = 1/2 or  23/4

please click onthe thanks button blue.. above...
Tanx!
U helped a lot!!