For the equation x^2 + kx + 9 = 0 to have real roots, what must be true about k?

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Question: For the equation x^2 + kx + 9 = 0 to have real roots, what must be true about k?

Options:

Correct Answer: k < 6

Solution:

The discriminant must be non-negative: k^2 - 4*1*9 >= 0 => k^2 >= 36 => k <= -6 or k >= 6.

For the equation x^2 + kx + 9 = 0 to have real roots, what must be true about k?

For the equation x^2 + kx + 9 = 0 to have real roots, what must be true about k?

Step 1: Identify the equation given, which is x^2 + kx + 9 = 0.
Step 2: Recognize that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: The discriminant formula for the equation ax^2 + bx + c is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = k, and c = 9.
Step 5: Substitute the values into the discriminant formula: D = k^2 - 4*1*9.
Step 6: Simplify the expression: D = k^2 - 36.
Step 7: Set the discriminant greater than or equal to zero for real roots: k^2 - 36 >= 0.
Step 8: Rearrange the inequality: k^2 >= 36.
Step 9: Solve for k by taking the square root: k >= 6 or k <= -6.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; for real roots, it must be non-negative.
Quadratic Formula – The quadratic formula is used to find the roots of a quadratic equation, and the discriminant is part of this formula.