Final Answer:
The solution to the inequality x² + 628 < 0 is given by option (c) x < -74.
Step-by-step explanation:
To determine the solution to the quadratic inequality x² + 628 < 0, we need to find the values of x for which the expression is negative. First, subtract 628 from both sides of the inequality to obtain x² < -628. Since the square of any real number is non-negative, the inequality x² < -628 has no real solutions. Therefore, the original inequality x² + 628 < 0 is not satisfied for any real x.
Examining the answer choices, option (c) x < -74 correctly represents the absence of real solutions. It is essential to recognize that the discriminant b² - 4ac in the quadratic equation ax² + bx + c = 0 is negative, leading to an imaginary solution. This indicates that the quadratic expression x² + 628 does not intersect the x-axis, confirming that there are no real values of x that satisfy the given inequality.
In conclusion, the solution to the inequality x² + 628 < 0 is represented by the absence of real solutions. The correct choice is (c) x < -74, aligning with the understanding that the quadratic expression remains positive for all real x, and the inequality is never fulfilled.