Find the value of k for which the roots of the equation x^2 - kx + 9 = 0 are rea

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Question: Find the value of k for which the roots of the equation x^2 - kx + 9 = 0 are real and distinct.

Options:

Correct Answer: k < 6

Solution:

The discriminant must be positive: k^2 - 4*1*9 > 0, which gives k < 6 or k > -6.

Find the value of k for which the roots of the equation x^2 - kx + 9 = 0 are real and distinct.

Find the value of k for which the roots of the equation x^2 - kx + 9 = 0 are real and distinct.

Step 1: Identify the quadratic equation given, which is x^2 - kx + 9 = 0.
Step 2: Recall 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 = -k, and c = 9.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-k)^2 - 4*1*9.
Step 5: Simplify the expression: D = k^2 - 36.
Step 6: For the roots to be real and distinct, the discriminant must be greater than 0: k^2 - 36 > 0.
Step 7: Solve the inequality k^2 - 36 > 0 by factoring: (k - 6)(k + 6) > 0.
Step 8: Determine the intervals where the product (k - 6)(k + 6) is positive. This occurs when k < -6 or k > 6.
Step 9: Conclude that the values of k for which the roots are real and distinct are k < -6 or k > 6.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; it must be positive for the roots to be real and distinct.
Quadratic Equation – A polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.