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

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

Options:

Correct Answer: -4

Solution:

For no real roots, the discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => |k| < 8.

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

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

Step 1: Identify the equation given, which is x^2 + kx + 16 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be less than zero.
Step 3: Write down 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 less than zero 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 can be any value between -8 and 8, but not including -8 and 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.
Inequalities – Understanding how to manipulate and solve inequalities is crucial for determining the range of values for k.