Question

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).

Answer 1

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

Answer 2

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