If x^2 - 4x + k has a double root, what is the value of k?

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Question: If x^2 - 4x + k has a double root, what is the value of k?

Options:

Correct Answer: 4

Solution:

For a double root, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(1)(k) = 0. Thus, 16 - 4k = 0, leading to k = 4.

If x^2 - 4x + k has a double root, what is the value of k?

If x^2 - 4x + k has a double root, what is the value of k?

Step 1: Identify the quadratic equation given, which is x^2 - 4x + k.
Step 2: Recognize that for a quadratic equation to have a double root, the discriminant must be zero.
Step 3: Write down the formula for the discriminant, which is b^2 - 4ac.
Step 4: Identify the coefficients from the equation: a = 1, b = -4, and c = k.
Step 5: Substitute the values of a, b, and c into the discriminant formula: (-4)^2 - 4(1)(k).
Step 6: Simplify the expression: 16 - 4k.
Step 7: Set the discriminant equal to zero for a double root: 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: Calculate the value: k = 4.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the condition for double roots.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the nature of the roots of a quadratic equation.