Final answer:
The coordinates of the vertex for the equation (x - 5)² - 6 = 0 are (5, -6). The solutions to the equation are x = 5 + √6 and x = 5 - √6.
Step-by-step explanation:
To solve the quadratic equation (x - 5)² - 6 = 0 and find the coordinates of the vertex, we first need to understand that the given equation is already in the vertex form of a quadratic equation, which is y = (x - h)² + k, where (h,k) is the vertex of the parabola.
By observing the equation, we can see that (h,k) for our equation is (5, -6). This gives us the coordinates of the vertex directly without any further calculation needed.
To solve the equation, we can make the equation equal to zero and then take the square root of both sides. Our equation, (x - 5)² - 6 = 0, simplifies to (x - 5)² = 6. Taking the square root gives us x - 5 = ±√6, which yields two solutions for x when solved: x = 5 + √6 and x = 5 - √6.