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 range of k?

Options:

Correct Answer: k < 9

Solution:

For real and distinct roots, the discriminant must be positive: 3^2 - 4*1*k > 0 => 9 - 4k > 0 => k < 9.

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

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

Step 1: Identify the equation given, which is x^2 + 3x + k = 0.
Step 2: Recognize that for a quadratic equation to have real and distinct roots, the discriminant must be greater than zero.
Step 3: Write down the formula for the discriminant, which is b^2 - 4ac. Here, a = 1, b = 3, and c = k.
Step 4: Substitute the values into the discriminant formula: 3^2 - 4*1*k.
Step 5: Simplify the expression: 9 - 4k.
Step 6: Set up the inequality for real and distinct roots: 9 - 4k > 0.
Step 7: Solve the inequality for k: First, subtract 9 from both sides: -4k > -9.
Step 8: Divide both sides by -4. Remember to flip the inequality sign: k < 9.
Step 9: Conclude that the range of k for the roots to be real and distinct is k < 9.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for real and distinct roots.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the nature of the roots of a quadratic equation.