(1/2x-5)^3+(1/2x+7)=(x+2)^2

To solve this equation, we will first expand the cube of the binomial (1/2x - 5)^3 using the binomial theorem:

(1/2x - 5)^3 = (1/2x)^3 - 3(1/2x)^2(5) + 3(1/2x)(5)^2 - 5^3
= (1/8x^3) - (15/4x^2) + (75/2x) - 125

Now, let's substitute this expansion back into the original equation:

(1/8x^3) - (15/4x^2) + (75/2x) - 125 + (1/2x + 7) = (x + 2)^2

Combining like terms, we have:

(1/8x^3) - (15/4x^2) + (75/2x) + (1/2x) - 118 = (x + 2)^2

Now, let's simplify the right side of the equation:

(x + 2)^2 = x^2 + 4x + 4

Substituting this back into the equation, we have:

(1/8x^3) - (15/4x^2) + (75/2x) + (1/2x) - 118 = x^2 + 4x + 4

To eliminate the fractions, let's multiply the entire equation by 8x:

x^3 - 30x^2 + 300x + 4x^2 - 64x - 944 = 8x^3 + 32x^2 + 32x

Now, let's combine like terms:

x^3 - 30x^2 + 4x^2 + 300x - 64x - 32x^2 - 32x - 8x^3 - 8x = 944

Simplifying further:

-7x^3 - 58x^2 + 204x - 8x = 944

-7x^3 - 66x^2 + 196x = 944

Now, let's move all terms to one side of the equation:

-7x^3 - 66x^2 + 196x - 944 = 0

This is a cubic equation, which can be difficult to solve algebraically. However, we can use numerical methods or a graphing calculator to find the approximate solutions.