For the equation x^2 + 4x + k = 0 to have real roots, what must be the condition

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Question: For the equation x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2023)

Options:

Correct Answer: k >= 16

Exam Year: 2023

Solution:

The discriminant must be non-negative: 4^2 - 4*1*k >= 0 leads to 16 - 4k >= 0, thus k <= 4.

For the equation x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2023)

For the equation x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2023)

Step 1: Identify the equation given, which is x^2 + 4x + k = 0.
Step 2: Recognize that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula 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: Calculate 4^2, which is 16, so we have D = 16 - 4k.
Step 7: Set the discriminant greater than or equal to zero for real roots: 16 - 4k >= 0.
Step 8: Rearrange the inequality to isolate k: 16 >= 4k.
Step 9: Divide both sides by 4: 4 >= k.
Step 10: Rewrite the condition: k must be less than or equal to 4.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; for real roots, it must be non-negative.
Quadratic Equation – Understanding the standard form of a quadratic equation and how to manipulate it to find conditions on coefficients.