Find the value of k for which the equation x^2 + kx + 16 = 0 has no real roots.

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Question: Find the value of k for which the equation x^2 + kx + 16 = 0 has no real roots.

Options:

Correct Answer: k < 8

Solution:

For no real roots, the discriminant must be less than 0: k^2 - 4*1*16 < 0, which gives k < 8.

Find the value of k for which the equation x^2 + kx + 16 = 0 has no real roots.

Find the value of k for which the equation x^2 + kx + 16 = 0 has no real roots.

Correct Answer: k < 8

Step 1: Understand that the equation is in the form of ax^2 + bx + c = 0, where a = 1, b = k, and c = 16.
Step 2: Recall that for a quadratic equation to have no real roots, the discriminant must be less than 0.
Step 3: The discriminant (D) is calculated using the formula D = b^2 - 4ac.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = k^2 - 4*1*16.
Step 5: Simplify the expression: D = k^2 - 64.
Step 6: Set up the inequality for no real roots: k^2 - 64 < 0.
Step 7: Rearrange the inequality: k^2 < 64.
Step 8: Take the square root of both sides: |k| < 8.
Step 9: This means k must be between -8 and 8: -8 < k < 8.
Step 10: Therefore, the value of k for which the equation has no real roots is k < 8.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots; if it is less than zero, the equation has no real roots.
Quadratic Equation – A quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.