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Question
Find the values of k for which the roots are real and equal in each of the following equation:
x2 - 2(5 + 2k)x + 3(7 + 10k) = 0
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Solution
The given quadric equation is x2 - 2(5 + 2k)x + 3(7 + 10k) = 0, and roots are real and equal
Then find the value of k.
Here, a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)
As we know that D = b2 - 4ac
Putting the value of a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)
= (-2(5 + 2k))2 - 4 x (1) x 3(7 + 10k)
= 4(25 + 20k + 4k2) - 12(7 + 10k)
= 100 + 80k + 16k2 - 84 - 120k
= 16 - 40k + 16k2
The given equation will have real and equal roots, if D = 0
Thus,
16 - 40k + 16k2 = 0
8(2k2 - 5k + 2) = 0
2k2 - 5k + 2 = 0
Now factorizing of the above equation
2k2 - 5k + 2 = 0
2k2 - 4k - k + 2 = 0
2k(k - 2) - 1(k - 2) = 0
(k - 2)(2k - 1) = 0
So, either
k - 2 = 0
k = 2
Or
2k - 1 = 0
2k = 1
k = 1/2
Therefore, the value of k = 2, 1/2.
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