For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct

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Question: For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct roots?

Options:

Correct Answer: -4

Solution:

The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.

For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct roots?

For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct roots?

Step 1: Identify the equation given, which is x^2 + kx + 16 = 0.
Step 2: Understand that for the equation to have real and distinct roots, the discriminant must be positive.
Step 3: Recall the formula for the discriminant, which is D = b^2 - 4ac. Here, a = 1, b = k, and c = 16.
Step 4: Substitute the values into the discriminant formula: D = k^2 - 4*1*16.
Step 5: Simplify the expression: D = k^2 - 64.
Step 6: Set the discriminant greater than zero for real and distinct roots: k^2 - 64 > 0.
Step 7: Solve the inequality: k^2 > 64.
Step 8: Take the square root of both sides: k > 8 or k < -8.
Step 9: Conclude that k can be any value greater than 8 or less than -8 for the roots to be real and distinct.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; it is calculated as b^2 - 4ac.
Quadratic Equation – A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0.
Real and Distinct Roots – For a quadratic equation to have real and distinct roots, the discriminant must be greater than zero.