If \(-\frac{1}{2}\) is one of the roots of the equation \(kx^2 - kx - 3 = 0\), then the value of k is:
(1) -4
(2) 2
(3) -2
(4) 4
Answer (2)
The value of k that makes − 2 1 a root of the equation k x 2 − k x − 3 = 0 is 4 . Therefore, the correct option is (4) 4.
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To find the value of k when − 2 1 is one of the roots of the quadratic equation k x 2 − k x − 3 = 0 , we can utilize the fact that if r is a root of a quadratic equation a x 2 + b x + c = 0 , then it must satisfy the equation. This means:
k ( − 2 1 ) 2 − k ( − 2 1 ) − 3 = 0
Step-by-step Calculation:
Calculate ( − 2 1 ) 2 :
( − 2 1 ) 2 = 4 1
Substitute the values into the equation:
k ⋅ 4 1 + 2 k − 3 = 0
Simplify the equation. Start by multiplying through by 4 to eliminate the fractions:
k + 2 k − 12 = 0
Combine like terms:
3 k − 12 = 0
Solve for k :
3 k = 12
k = 3 12
k = 4
Thus, the value of k is 4 . Therefore, the correct option is (4) 4.
