If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the value of k?

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Q: If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the value of k?

Solution: For real and equal roots, the discriminant must be zero: 25 - 4k = 0, thus k = 6.

Steps: 10

Step 1: Identify the equation given, which is x^2 - 5x + k = 0.
Step 2: Understand that for the roots of a quadratic equation to be real and equal, the discriminant must be zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is calculated using the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -5, and c = k.
Step 5: Substitute the values into the discriminant formula: D = (-5)^2 - 4(1)(k).
Step 6: Simplify the expression: D = 25 - 4k.
Step 7: Set the discriminant equal to zero for real and equal roots: 25 - 4k = 0.
Step 8: Solve for k by rearranging the equation: 4k = 25.
Step 9: Divide both sides by 4 to find k: k = 25 / 4.
Step 10: Calculate the value: k = 6.25.