The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?

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Question: The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?

Options:

Correct Answer: 8

Solution:

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.

The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?

The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?

Step 1: Identify the equation given, which is x^2 + 4x + k = 0.
Step 2: Understand that for the roots of a quadratic equation to be equal, the discriminant must be zero.
Step 3: Recall the formula for the discriminant, which is b^2 - 4ac, where a, b, and c are the coefficients from the equation ax^2 + bx + c = 0.
Step 4: In our equation, a = 1, b = 4, and c = k.
Step 5: Substitute the values into the discriminant formula: 4^2 - 4*1*k.
Step 6: Calculate 4^2, which is 16, so we have 16 - 4k.
Step 7: Set the discriminant equal to zero for the roots to be equal: 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.

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 equal, D must be zero.