1. **Problem statement:** We have a right triangle with a vertical leg divided into two segments: the upper segment labeled $x$ and the lower segment labeled 2, the horizontal leg labeled 3, and the hypotenuse labeled 5. We need to find the value of $x$. 2. **Formula and rules:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Set up the equation:** The vertical leg is the sum of $x$ and 2, so its length is $x + 2$. The horizontal leg is 3, and the hypotenuse is 5. Applying the Pythagorean theorem: $$ (x + 2)^2 + 3^2 = 5^2 $$ 4. **Expand and simplify:** $$ (x + 2)^2 + 9 = 25 $$ $$ x^2 + 4x + 4 + 9 = 25 $$ $$ x^2 + 4x + 13 = 25 $$ 5. **Isolate terms:** $$ x^2 + 4x + 13 - 25 = 0 $$ $$ x^2 + 4x - 12 = 0 $$ 6. **Solve quadratic equation:** Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=4$, and $c=-12$. 7. **Calculate discriminant:** $$ \sqrt{4^2 - 4 \times 1 \times (-12)} = \sqrt{16 + 48} = \sqrt{64} = 8 $$ 8. **Find roots:** $$ x = \frac{-4 \pm 8}{2} $$ 9. **Evaluate each root:** - $$ x = \frac{-4 + 8}{2} = \frac{4}{2} = 2 $$ - $$ x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6 $$ 10. **Interpretation:** Since $x$ represents a length, it must be positive. Therefore, the solution is: $$ \boxed{2} $$