For which value of k does the equation x^2 + kx + 16 = 0 have no real roots?

For which value of k does the equation x^2 + kx + 16 = 0 have no real roots?

Step 1: Identify the equation given, which is x^2 + kx + 16 = 0.
Step 2: Recognize that to find when this equation has no real roots, we need to use the discriminant.
Step 3: The discriminant formula for a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = k, and c = 16.
Step 5: Substitute these values into the discriminant formula: D = k^2 - 4*1*16.
Step 6: Simplify the expression: D = k^2 - 64.
Step 7: For the equation to have no real roots, the discriminant must be less than zero: k^2 - 64 < 0.
Step 8: Rearrange the inequality: k^2 < 64.
Step 9: Take the square root of both sides: -8 < k < 8.
Step 10: Conclude that k must be between -8 and 8 (not including -8 and 8) for the equation to have no real roots.

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 Inequalities – Understanding how to manipulate inequalities to find ranges of values for parameters in quadratic equations.