Final answer:
To solve the quadratic equation x² - 6x + 1 = -7, we adjust it to x² - 6x + 8 = 0 and use the quadratic formula x = (-b ± √(b² - 4ac)) / (2a), because the equation does not simply factor into rational numbers.
Step-by-step explanation:
To solve the quadratic equation x² - 6x + 1 = -7, we start by adding 7 to both sides of the equation to set it to zero: x² - 6x + 8 = 0. After that, we need to factor the left-hand side of the equation. Unfortunately, this equation does not factor neatly into rational numbers, so we'll need to use the quadratic formula to find the roots.
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are coefficients from the equation ax² + bx + c = 0. For our equation, a=1, b=-6, and c=8. Applying these values to the quadratic formula will give us the solutions for x.