For the equation x^2 + 6x + k = 0 to have no real roots, what must be the condit

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Question: For the equation x^2 + 6x + k = 0 to have no real roots, what must be the condition on k?

Options:

Correct Answer: k ≤ 0

Solution:

The condition for no real roots is that the discriminant must be less than zero: 6^2 - 4*1*k < 0 => 36 < 4k => k > 9.

For the equation x^2 + 6x + k = 0 to have no real roots, what must be the condition on k?

For the equation x^2 + 6x + k = 0 to have no real roots, what must be the condition on k?

Step 1: Identify the equation given, which is x^2 + 6x + k = 0.
Step 2: Recognize that for a quadratic equation to have no real roots, the discriminant must be less than zero.
Step 3: Write down the formula for the discriminant, which is D = b^2 - 4ac. Here, a = 1, b = 6, and c = k.
Step 4: Substitute the values into the discriminant formula: D = 6^2 - 4*1*k.
Step 5: Simplify the expression: D = 36 - 4k.
Step 6: Set the discriminant less than zero for no real roots: 36 - 4k < 0.
Step 7: Rearrange the inequality: 36 < 4k.
Step 8: Divide both sides by 4 to isolate k: 9 < k.
Step 9: Write the final condition: k must be greater than 9.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; if it is less than zero, the equation has no real roots.
Quadratic Equation – A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.