To solve the equation $x^3 - 5x^2 + x + 15 = 0$ graphically, follow these steps:

1. Graph the function: Input $f(x) = x^3 - 5x^2 + x + 15$ into a graphing calculator.
2. Identify the x-intercepts: The real solutions of the equation correspond to the x-intercepts of the graph, i.e., where $f(x) = 0$.
3. Locate x-intercepts: Use the calculator's "zero" or "root" function to pinpoint the x-values where the curve crosses the x-axis.
4. Check each solution to ensure accuracy to at least two decimal places.

Graphical Analysis Result

The graph reveals three real solutions at the following points:

1. $x \approx -2.69$
2. $x \approx 0.88$
3. $x \approx 6.81$

These values are accurate to two decimal places.

Would you like a detailed explanation or have any questions?

Related Questions

1. What are the characteristics of cubic equations with three distinct real roots?
2. How can you verify the roots using synthetic division?
3. What role does the derivative $f'(x)$ play in finding critical points for the curve?
4. How do turning points relate to the roots of cubic functions?
5. What are some alternative methods to graphically solving polynomial equations?

Tip

Always adjust your graphing window to include regions where the curve intersects the x-axis, ensuring all roots are visible.