If the roots of the equation x^2 + 6x + k = 0 are real and distinct, what must b

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Question: If the roots of the equation x^2 + 6x + k = 0 are real and distinct, what must be the condition on k? (2023)

Options:

Correct Answer: k < 9

Exam Year: 2023

Solution:

For real and distinct roots, the discriminant must be greater than zero: 6^2 - 4*1*k > 0 leads to k < 9.

If the roots of the equation x^2 + 6x + k = 0 are real and distinct, what must be the condition on k? (2023)

If the roots of the equation x^2 + 6x + k = 0 are real and distinct, what must be the condition on k? (2023)

Step 1: Identify the equation given, which is x^2 + 6x + k = 0.
Step 2: Recognize 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: Simplify the expression: D = 36 - 4k.
Step 6: For the roots to be real and distinct, the discriminant must be greater than zero: 36 - 4k > 0.
Step 7: Rearrange the inequality: 36 > 4k.
Step 8: Divide both sides by 4 to isolate k: 9 > k.
Step 9: Rewrite the condition: k must be less than 9.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots. For real and distinct roots, the discriminant must be greater than zero.
Quadratic Equation – A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Inequalities – Understanding how to manipulate inequalities is crucial for determining the conditions on k.