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

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

Options:

Correct Answer: 6

Solution:

The minimum value of k is 6, as the discriminant must be zero.

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

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

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

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. For the roots to be real and equal, D must be equal to zero.
Quadratic Equations – Understanding the properties of quadratic equations, including the conditions for real and equal roots.