Solve (x-1)2 = [4√(x-4)]2

 To solve this problem, we need to use the properties of exponents and square roots. First, we can simplify the left side of the equation by squaring the quantity inside the parentheses:

(x-1)^2 = [4√(x-4)]^2

Now, let's move the square root on the right side of the equation inside the square, by taking the square root of both sides of the equation:

(x-1)^2 = 4^2 * (x-4)

Next, we can simplify the right side of the equation by using the properties of exponents. Since 4^2 is the same as 4 * 4, we can rewrite the right side of the equation as:

(x-1)^2 = 4 * 4 * (x-4)

Now we can distribute the 4 to get:

(x-1)^2 = 16 * (x-4)

Finally, we can solve the equation by setting the left and right sides of the equation equal to each other and then solving for x:

(x-1)^2 = 16 * (x-4)
x^2 - 2x + 1 = 16x - 64
x^2 - 18x + 65 = 0

To solve for x, we can use the quadratic formula, which states that the solutions to a quadratic equation in the form ax^2 + bx + c = 0 are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a is 1, b is -18, and c is 65. Plugging these values into the formula, we get:

x = (18 ± √(18^2 - 4 * 1 * 65)) / 2 * 1

We can simplify this to get:

x = (18 ± √(324 - 260)) / 2

And then:

x = (18 ± √(64)) / 2

The square root of 64 is 8, so we can compute:

x = (18 + 8) / 2 or (18 - 8) / 2

This gives us:

x = 26 / 2 or x = 10 / 2

Finally, we can simplify these expressions to get the solutions:

x = 13 or x = 5

Therefore, the solutions to the equation (x-1)^2 = [4√(x-4)]^2 are x = 13 and x = 5.