Solution:

Given: x = 1/2 - √3

To find the value of x^3 - 2x^2 - 7x + 5, we need to substitute the value of x in the given expression and simplify it step by step.

Step 1: Substitute the value of x in the expression x^3 - 2x^2 - 7x + 5:

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

Step 2: Simplify the expression:

To simplify the expression, we need to expand the cube of (1/2 - √3) and square of (1/2 - √3).

Expanding the cube of (1/2 - √3):
(1/2 - √3)^3 = (1/2)^3 - 3(1/2)^2(√3) + 3(1/2)(√3)^2 - (√3)^3
= 1/8 - 3/4(√3) + 3/2(3) - 3√3
= 1/8 - 3/4√3 + 9/2 - 3√3
= 9/2 - 7/4√3

Expanding the square of (1/2 - √3):
(1/2 - √3)^2 = (1/2)^2 - 2(1/2)(√3) + (√3)^2
= 1/4 - √3 + 3
= 10/4 - √3
= 5/2 - √3

Substituting these values back in the original expression:

x^3 - 2x^2 - 7x + 5 = (9/2 - 7/4√3) - 2(5/2 - √3) - 7(1/2 - √3) + 5

Step 3: Simplify further:

Simplifying the expression, we get:

(9/2 - 7/4√3) - 2(5/2 - √3) - 7(1/2 - √3) + 5
= 9/2 - 7/4√3 - 5 + 2√3 - 7/2 + 7√3 + 5
= -3/2 + (2√3 - 7/4√3 + 7√3) + 9/2

Step 4: Combine like terms:

Combining the like terms, we have:

-3/2 + (2√3 - 7/4√3 + 7√3) + 9/2
= -3/2 + (2 + 7 - 7/4)√3 + 9/2
= -3/