Use a graphing calculator to solve the equation by graphing. If necessary, round to the nearest hundredth. \( x^{3}-9 x^{2}+27 x=20 \)

To solve the equation \( x^3 - 9x^2 + 27x = 20 \) using a graphing calculator, we'll follow these steps: 1. **Rearrange the Equation:** First, rewrite the equation in the standard form: \[ x^3 - 9x^2 + 27x - 20 = 0 \] 2. **Graph the Function:** Plot the function \( f(x) = x^3 - 9x^2 + 27x - 20 \) on a graphing calculator. 3. **Identify the Root:** By observing the graph, you'll notice that the function crosses the x-axis between \( x = 1 \) and \( x = 2 \). This indicates there's a real root in this interval. 4. **Narrow Down the Root:** To approximate the root more precisely: - **At \( x = 1.08 \):** \[ f(1.08) \approx -0.0779 \] - **At \( x = 1.09 \):** \[ f(1.09) \approx 0.0322 \] The root lies between \( x = 1.08 \) and \( x = 1.09 \). 5. **Approximate the Root:** Using linear interpolation between these points: \[ x \approx 1.087 \] Rounding to the nearest hundredth: \[ x \approx 1.09 \] **Solution:** The solution to the equation \( x^3 - 9x^2 + 27x = 20 \) is approximately: \[ x \approx 1.09 \]