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Question
The positive value of k for which the equation x2 + kx + 64 = 0 and x2 – 8x + k = 0 will both have real roots, is ______.
Options
4
8
12
16
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Solution
The positive value of k for which the equation x2 + kx + 64 = 0 and x2 – 8x + k = 0 will both have real roots, is 16.
Explanation:
The given quadric equation are x2 + kx + 64 = 0 and x2 – 8x + k = 0 roots are real.
Then find the value of a.
Here, x2 + kx + 64 = 0 ...(1)
x2 – 8x + k = 0 ...(2)
a1 = 1, b1 = k and c1 = 64
a2 = 1, b2 = –8 and c2 = k
As we know that D1 = b2 – 4ac
Putting the value of a1 = 1, b1 = k and c1 = 64
= (k)2 – 4 × 1 × 64
= k2 – 256
The given equation will have real and distinct roots, if D > 0
k2 – 256 = 0
k2 = 256
`k = sqrt256`
k = ±16
Therefore, putting the value of k = 16 in equation (2), we get
x2 – 8x + 16 = 0
(x – 4)2 = 0
x – 4 = 0
x = 4
The value of k = 16 satisfying to both equations.
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