Use a graphing calculator to solve the equation by graphing. If necessary, round to the nearest hundredth.

Question

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

UpStudy AI · Accepted Answer

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:
$$
x \approx 1.09
$$
The solution to the equation $$ x^3 - 9x^2 + 27x = 20 $$ is approximately $$ x \approx 1.09 $$.

Use a graphing calculator to solve the equation by graphing. If necessary, round to the nearest hundredth.

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