If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the

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Question: If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the condition on k? (2022)

Options:

Correct Answer: k < 9

Exam Year: 2022

Solution:

The discriminant must be positive: 3^2 - 4*1*k > 0 leads to 9 - 4k > 0, thus k < 9.

If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the condition on k? (2022)

If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the condition on k? (2022)

Step 1: Identify the equation given, which is x^2 + 3x + k = 0.
Step 2: Recognize that for the roots of a quadratic equation to be real and distinct, the discriminant must be positive.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = 3, and c = k. So, we calculate the discriminant: D = 3^2 - 4*1*k.
Step 5: Simplify the discriminant: D = 9 - 4k.
Step 6: Set the condition for the roots to be real and distinct: 9 - 4k > 0.
Step 7: Solve the inequality: 9 > 4k, which can be rewritten as k < 9.
Step 8: Conclude that the condition on k for the roots to be real and distinct is k < 9.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. It determines the nature of the roots: if D > 0, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, the roots are complex.
Conditions for Real and Distinct Roots – For the roots of a quadratic equation to be real and distinct, the discriminant must be positive.