Final answer:
To solve the equation x² + 6x = -25, one can use the quadratic formula, which results in the complex solutions x = -3 + 4i and x = -3 - 4i.
Step-by-step explanation:
The equation in question is x² + 6x = -25. To solve this equation, we need to rearrange it into a standard quadratic form, which is ax² + bx + c = 0. Adding 25 to both sides of the equation gives us x² + 6x + 25 = 0. This equation does not factor neatly, so we employ the quadratic formula:
x = √-(-b ± √(b² - 4ac)) / (2a)
Here, a = 1, b = 6, and c = 25. Plugging these values into the formula yields:
x = (-6 ± √(6² - 4(1)(25))) / (2(1))
x = (-6 ± √(36 - 100)) / 2
x = (-6 ± √(-64)) / 2
Since we have a negative number under the square root, this will result in complex numbers. The square root of -64 is 8i, where i is the imaginary unit. Therefore, the solutions are:
x = (-6 + 8i) / 2 and x = (-6 - 8i) / 2,
which simplifies to x = -3 + 4i and x = -3 - 4i.