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Question
If alpha beta are the roots of quadratic polynomial p(X)=xsquare -(k+6)X+2(2k-1) find the value of k when alpha+beta = 1alphabeta/2
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Solution
Let α and β are zeroes of the polynomial = x² - (k + 6)x + 2(2k –1).
On comparing with ax²+bx+c=0
a= 1, b= -( k+6) , c = 2(2k –1)
Sum of zeroes (α+β)= -b/a = -(-(k+6))/1
α+β= (k+6)…………....(1)
Product of zeros(α.β)= c/a = 2(2k –1)/1
α.β= c/a = 4k -2…………(2)
Given: (α+β) = ½(αβ )
(k+6) = ½( 4k -2)
[From eq 1 & 2]
2 (k +6 )= 4k -2
2k +12 = 4k -2
2k -4k = -2 -12
-2k = -14
k = 14/2
k = 7
Hence, the value of k is 7
On comparing with ax²+bx+c=0
a= 1, b= -( k+6) , c = 2(2k –1)
Sum of zeroes (α+β)= -b/a = -(-(k+6))/1
α+β= (k+6)…………....(1)
Product of zeros(α.β)= c/a = 2(2k –1)/1
α.β= c/a = 4k -2…………(2)
Given: (α+β) = ½(αβ )
(k+6) = ½( 4k -2)
[From eq 1 & 2]
2 (k +6 )= 4k -2
2k +12 = 4k -2
2k -4k = -2 -12
-2k = -14
k = 14/2
k = 7
Hence, the value of k is 7
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