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

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

For the quadratic equation x^2 + 6x + k = 0 to have real 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 quadratic to have real roots, the discriminant must be non-negative. This means D ≥ 0.
Step 7: Set up the inequality: 36 - 4k ≥ 0.
Step 8: Rearrange the inequality to isolate k: 36 ≥ 4k.
Step 9: Divide both sides by 4 to solve for k: 9 ≥ k.
Step 10: Rewrite the result: k must be less than or equal to 9, or k ≤ 9.

Quadratic Equations – Understanding the conditions for real roots using the discriminant.
Discriminant – The formula used to determine the nature of the roots of a quadratic equation.