Final answer:
The equation x² + 25 = 0 has no real solutions and yields two complex roots, which are x = 5i and x = -5i. These roots are complex numbers because the equation involves the square root of a negative number.
Step-by-step explanation:
The equation x² + 25 = 0 does not have real solutions because the sum of a square (which is always non-negative) and a positive number cannot equal zero. To solve the equation, we could attempt to use the quadratic formula:
ax² + bx + c = 0
However, in this equation, a = 1, b = 0, and c = 25. The quadratic formula requires calculating the discriminant (b² - 4ac), which in this case is (0² - 4(1)(25)) = -100. Since the discriminant is negative, this indicates that there are no real solutions, but rather two complex roots which are conjugates of each other. These complex numbers can be expressed as:
- x = ( -b + √(b² - 4ac) ) / (2a)
- x = ( -b - √(b² - 4ac) ) / (2a)
Substituting into the formulas:
- x = ( -0 + √(0 - 4(1)(25)) ) / (2(1))
- x = ( -0 - √(0 - 4(1)(25)) ) / (2(1))
The solutions are x = 5i and x = -5i, where i is the imaginary unit (i² = -1). So, the types of numbers that the solutions represent are complex numbers.