If the roots of the equation x^2 + 4x + k = 0 are real and distinct, what is the condition on k?

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Q: If the roots of the equation x^2 + 4x + k = 0 are real and distinct, what is the condition on k?

Solution: The discriminant must be greater than zero: 4^2 - 4*1*k > 0 => 16 - 4k > 0 => k < 4.

Steps: 9

Step 1: Identify the equation given, which is x^2 + 4x + k = 0.
Step 2: Recognize 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 ax^2 + bx + c = 0 is calculated using the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = 4, and c = k. So, we substitute these values into the discriminant formula: D = 4^2 - 4*1*k.
Step 5: Calculate 4^2, which is 16. Now we have D = 16 - 4k.
Step 6: Set the discriminant greater than zero for the roots to be real and distinct: 16 - 4k > 0.
Step 7: Solve the inequality 16 - 4k > 0. First, subtract 16 from both sides: -4k > -16.
Step 8: Divide both sides by -4. Remember, when dividing by a negative number, the inequality sign flips: k < 4.
Step 9: Conclude that the condition on k for the roots to be real and distinct is k < 4.