What is the value of k if the equation x^2 + kx + 16 = 0 has no real roots?
Correct Answer: k < -8 or k > 8
- 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: The discriminant formula for the equation ax^2 + bx + c is given by D = b^2 - 4ac.
- Step 4: In our equation, a = 1, b = k, and c = 16.
- Step 5: Substitute the values into the discriminant formula: D = k^2 - 4*1*16.
- Step 6: Simplify the expression: D = k^2 - 64.
- Step 7: Set the discriminant less than zero for no real roots: k^2 - 64 < 0.
- Step 8: Rearrange the inequality: k^2 < 64.
- Step 9: Take the square root of both sides: |k| < 8.
- Step 10: This means k can be any value between -8 and 8, but not including -8 and 8.
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