The equation x^2 - 4x + k = 0 has equal roots when k is equal to:

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Question: The equation x^2 - 4x + k = 0 has equal roots when k is equal to:

Options:

Correct Answer: 4

Solution:

For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0 leads to k = 4.

The equation x^2 - 4x + k = 0 has equal roots when k is equal to:

The equation x^2 - 4x + k = 0 has equal roots when k is equal to:

Step 1: Identify the equation given, which is x^2 - 4x + k = 0.
Step 2: Recognize that for the equation to have equal roots, the discriminant must be zero.
Step 3: The discriminant formula for a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -4, and c = k.
Step 5: Substitute the values into the discriminant formula: D = (-4)^2 - 4*1*k.
Step 6: Simplify the expression: D = 16 - 4k.
Step 7: Set the discriminant equal to zero for equal roots: 16 - 4k = 0.
Step 8: Solve for k by adding 4k to both sides: 16 = 4k.
Step 9: Divide both sides by 4 to isolate k: k = 16 / 4.
Step 10: Simplify the division: k = 4.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; for equal roots, it must be zero.
Quadratic Equation – A quadratic equation is in the form ax^2 + bx + c = 0, where a, b, and c are constants.