Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no re

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Question: Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no real roots. (2020)

Options:

Correct Answer: -8

Exam Year: 2020

Solution:

The discriminant must be less than zero: k^2 - 4*1*16 < 0 leads to k < -8.

Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no real roots. (2020)

Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no real roots. (2020)

Step 1: Identify the quadratic equation, which is x^2 + kx + 16 = 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 = 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: For the quadratic equation to have no real roots, the discriminant must be less than zero: k^2 - 64 < 0.
Step 7: Rearrange the inequality: k^2 < 64.
Step 8: Take the square root of both sides: -8 < k < 8.
Step 9: Since we want the value of k for which there are no real roots, we focus on the part where k < -8.

Quadratic Equations – Understanding the conditions under which a quadratic equation has real or complex roots, specifically using the discriminant.
Discriminant – The discriminant (D = b^2 - 4ac) determines the nature of the roots of a quadratic equation.