1. **Stating the problem:** We have a triangle with sides given by expressions $20 - 2x$, $\frac{1}{2}x + 10$, and $x^2 - x$. We need to find the value of $x$ for which the triangle is equilateral, meaning all sides are equal. 2. **Formula and rule:** For an equilateral triangle, all three sides must be equal: $$20 - 2x = \frac{1}{2}x + 10 = x^2 - x$$ 3. **Step 1: Equate the first two sides:** $$20 - 2x = \frac{1}{2}x + 10$$ 4. **Solve for $x$:** $$20 - 2x = \frac{1}{2}x + 10$$ $$20 - 10 = \frac{1}{2}x + 2x$$ $$10 = \frac{1}{2}x + 2x$$ $$10 = \frac{1}{2}x + \frac{4}{2}x = \frac{5}{2}x$$ $$x = \frac{10}{\frac{5}{2}} = 10 \times \frac{2}{5} = 4$$ 5. **Step 2: Check if $x=4$ satisfies the third side equality:** Calculate each side: $$20 - 2(4) = 20 - 8 = 12$$ $$\frac{1}{2}(4) + 10 = 2 + 10 = 12$$ $$4^2 - 4 = 16 - 4 = 12$$ All sides equal 12, so $x=4$ makes the triangle equilateral. 6. **Check other options (optional):** - For $x=3.5$, sides are $20 - 7 = 13$, $1.75 + 10 = 11.75$, $12.25 - 3.5 = 8.75$ (not equal). - For $x=2.5$, sides are $20 - 5 = 15$, $1.25 + 10 = 11.25$, $6.25 - 2.5 = 3.75$ (not equal). - For $x=5$, sides are $20 - 10 = 10$, $2.5 + 10 = 12.5$, $25 - 5 = 20$ (not equal). **Final answer:** $\boxed{4}$