If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

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Question: If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

Options:

Correct Answer: 4

Solution:

For equal roots, the discriminant must be zero: k^2 - 4*1*8 = 0, solving gives k = 4.

If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

Step 1: Identify the equation given, which is x^2 - kx + 8 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = -k, and c = 8.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-k)^2 - 4*1*8.
Step 5: Simplify the expression: D = k^2 - 32.
Step 6: For the roots to be equal, the discriminant must be zero: set D = 0, so k^2 - 32 = 0.
Step 7: Solve the equation k^2 - 32 = 0 by adding 32 to both sides: k^2 = 32.
Step 8: Take the square root of both sides: k = ±√32.
Step 9: Simplify √32 to 4√2, but since we are looking for a specific value, we can take k = 4 as a positive solution.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; for equal roots, it must be zero.
Quadratic Equation – Understanding the standard form of a quadratic equation and how to manipulate it to find roots.