For the equation x² + 6x + k = 0 to have real roots, what is the minimum value o

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Question: For the equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)

Options:

Correct Answer: -9

Solution:

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, which gives k ≤ 9, so the minimum value is -9.

For the equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)

For the equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)

Step 1: Identify the equation given, which is x² + 6x + k = 0.
Step 2: Recognize that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: Write down the formula for the discriminant, which is D = b² - 4ac. Here, a = 1, b = 6, and c = k.
Step 4: Substitute the values into the discriminant formula: D = 6² - 4*1*k.
Step 5: Simplify the expression: D = 36 - 4k.
Step 6: Set the discriminant greater than or equal to zero for real roots: 36 - 4k ≥ 0.
Step 7: Rearrange the inequality to solve for k: 36 ≥ 4k.
Step 8: Divide both sides by 4: 9 ≥ k.
Step 9: This means k must be less than or equal to 9 for the equation to have real roots.
Step 10: The minimum value of k that satisfies this condition is -9.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; for real roots, it must be non-negative.
Quadratic Inequality – Understanding how to manipulate inequalities to find the range of values for parameters in quadratic equations.