If \(\dfrac{1}{2}\) is a root of \(x^2+kx-\dfrac{5}{4}=0\), the value of \(k\) is
2
−2
\(\dfrac{1}{4}\)
\(\dfrac{1}{2}\)
We are told that \(\tfrac{1}{2}\) is a root of the quadratic equation:
\(x^2 + kx - \tfrac{5}{4} = 0\)
Step 1: If a number is a root of the equation, it means that if we put that number in place of \(x\), the whole equation will become zero.
Step 2: Put \(x = \tfrac{1}{2}\) in the equation:
\(\left(\tfrac{1}{2}\right)^2 + k\left(\tfrac{1}{2}\right) - \tfrac{5}{4} = 0\)
Step 3: Simplify each term:
So the equation becomes:
\(\tfrac{1}{4} + \tfrac{k}{2} - \tfrac{5}{4} = 0\)
Step 4: Combine the fractions \(\tfrac{1}{4} - \tfrac{5}{4}\):
\(\tfrac{1 - 5}{4} = -\tfrac{4}{4} = -1\)
Now the equation looks like:
\(\tfrac{k}{2} - 1 = 0\)
Step 5: Add 1 to both sides:
\(\tfrac{k}{2} = 1\)
Step 6: Multiply both sides by 2 to find \(k\):
\(k = 2\)
Final Answer: \(k = 2\)