X+24= (x-3)√(-x^2+8x+48)

To solve the equation x + 24 = (x - 3)√(-x^2 + 8x + 48), we need to isolate the radical term on one side of the equation and square both sides to eliminate the square root.

First, let's move the x term to the right side of the equation:

24 = (x - 3)√(-x^2 + 8x + 48) - x

Next, square both sides of the equation:

576 = [(x - 3)√(-x^2 + 8x + 48) - x]^2

Expanding the square on the right side:

576 = (x - 3)^2(-x^2 + 8x + 48) - 2x(x - 3)√(-x^2 + 8x + 48) + x^2

Simplifying:

576 = (x^2 - 6x + 9)(-x^2 + 8x + 48) - 2x(x - 3)√(-x^2 + 8x + 48) + x^2

576 = -x^4 + 8x^3 + 48x^2 - 6x^3 + 48x^2 - 288x + 9x^2 - 72x + 432 - 2x(x - 3)√(-x^2 + 8x + 48) + x^2

Combining like terms:

576 = -x^4 + 2x^3 + 56x^2 - 360x + 432 - 2x(x - 3)√(-x^2 + 8x + 48)

Now, let's isolate the radical term:

2x(x - 3)√(-x^2 + 8x + 48) = -x^4 + 2x^3 + 56x^2 - 360x + 432 - 576

2x(x - 3)√(-x^2 + 8x + 48) = -x^4 + 2x^3 + 56x^2 - 360x - 144

Now, square both sides again to eliminate the square root:

[2x(x - 3)√(-x^2 + 8x + 48)]^2 = (-x^4 + 2x^3 + 56x^2 - 360x - 144)^2

4x^2(x - 3)^2(-x^2 + 8x + 48) = (-x^4 + 2x^3 + 56x^2 - 360x - 144)^2

Expanding the square on the right side:

4x^2(x - 3)^2(-x^2 + 8x + 48) = x^8 - 4x^7 + 60x^6 - 400x^5 + 1152x^4 - 1728x^3 + 1296x^2 - 576x + 20736

Simplifying:

4x^2(x - 3)^2(-x^2 + 8x + 48) - x^8 + 4x^7 - 60x^6 + 400x^5 - 1152x^4 + 1728x^3 - 1296x^2 + 576x - 20736 = 0

This is a polynomial equation of degree 8. Unfortunately, it is not possible to solve this equation algebraically. You would need to use numerical methods or a graphing calculator to approximate the solutions.