If the roots of the equation x² + 5x + k = 0 are real and distinct, what is the condition for k? (2020)

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Q: If the roots of the equation x² + 5x + k = 0 are real and distinct, what is the condition for k? (2020)

Solution: The discriminant must be greater than zero: 5² - 4*1*k > 0, thus k < 25.

Steps: 9

Step 1: Identify the equation given, which is x² + 5x + k = 0.
Step 2: Understand that for the roots of a quadratic equation to be real and distinct, the discriminant must be greater than zero.
Step 3: The discriminant (D) for the equation ax² + bx + c = 0 is calculated using the formula D = b² - 4ac.
Step 4: In our equation, a = 1, b = 5, and c = k. So, we substitute these values into the discriminant formula: D = 5² - 4*1*k.
Step 5: Calculate 5², which is 25. Now we have D = 25 - 4k.
Step 6: Set up the inequality for the discriminant to be greater than zero: 25 - 4k > 0.
Step 7: Solve the inequality for k. First, rearrange it: 25 > 4k.
Step 8: Divide both sides by 4 to isolate k: 25/4 > k, or k < 25/4.
Step 9: Since 25/4 equals 6.25, we can say k must be less than 6.25 for the roots to be real and distinct.