For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be

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Question: For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)

Options:

Correct Answer: k < 9

Exam Year: 2020

Solution:

The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.

For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)

For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)

Step 1: Identify the quadratic equation, which is x^2 + 6x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = 6, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = 6^2 - 4*1*k.
Step 5: Calculate 6^2, which is 36. So, D = 36 - 4k.
Step 6: For the roots to be distinct, the discriminant must be greater than 0: 36 - 4k > 0.
Step 7: Rearrange the inequality: 36 > 4k.
Step 8: Divide both sides by 4 to isolate k: 9 > k.
Step 9: Rewrite the condition: k must be less than 9.

Quadratic Equations – Understanding the conditions for distinct roots using the discriminant.
Discriminant – The formula (b^2 - 4ac) used to determine the nature of the roots of a quadratic equation.