For the quadratic equation x² + 6x + k = 0 to have real roots, what is the minim

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

Options:

Correct Answer: -9

Exam Year: 2021

Solution:

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9. The minimum value of k is -9.

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

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

Step 1: Identify the quadratic equation, which is x² + 6x + k = 0.
Step 2: Recall that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: The discriminant (D) for the equation ax² + bx + c = 0 is given by the formula D = b² - 4ac.
Step 4: In our equation, a = 1, b = 6, and c = k. So, we calculate the discriminant: D = 6² - 4*1*k.
Step 5: Simplify the discriminant: 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: 36 ≥ 4k.
Step 8: Divide both sides by 4: 9 ≥ k.
Step 9: This means k must be less than or equal to 9: k ≤ 9.
Step 10: To find the minimum value of k, we consider the lowest value that k can take while still allowing real roots, which is k = -9.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. It determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.
Condition for Real Roots – For a quadratic equation to have real roots, the discriminant must be non-negative (D ≥ 0).
Finding Minimum Value – To find the minimum value of k for which the quadratic has real roots, we rearrange the inequality derived from the discriminant.