1. **State the problem:** We are given the function $f(x) = x^2 - \sqrt{x}$ and need to find the equation of the tangent line at $a = 1$. 2. **Recall the formula for the tangent line:** The equation of the tangent line to $f(x)$ at $x = a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f(x)$ evaluated at $x = a$. 3. **Find $f(a)$:** $$f(1) = 1^2 - \sqrt{1} = 1 - 1 = 0$$ 4. **Find the derivative $f'(x)$:** $$f(x) = x^2 - x^{1/2}$$ Using power rule: $$f'(x) = 2x - \frac{1}{2}x^{-1/2}$$ 5. **Evaluate $f'(a)$ at $a=1$:** $$f'(1) = 2(1) - \frac{1}{2}(1)^{-1/2} = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}$$ 6. **Write the tangent line equation:** $$y = f(1) + f'(1)(x - 1) = 0 + \frac{3}{2}(x - 1) = \frac{3}{2}x - \frac{3}{2}$$ **Final answer:** The equation of the tangent line at $x=1$ is: $$y = \frac{3}{2}x - \frac{3}{2}$$