I see you've uploaded an image with four cubic equations and their graphs. Let me process the content of this image to solve the equations and analyze the graphs. I'll also provide a conjecture on determining the number and types of solutions for cubic functions based on the graphs. PIL import Image import pytesseract

Load the uploaded image

image_path = '/mnt/data/1736891832770cped93zq.jpg' img = Image.open(image_path)

Use OCR to extract text

extracted_text = pytesseract.image_to_string(img) extracted_textThe OCR output was partially successful but includes errors and missing information. I'll manually refine the equations and extract the necessary data based on the visible content in the image.

From the table:

1. The cubic equations are clearly provided in the first column.
2. The graphs help identify the real and complex roots visually.

Let me proceed to solve the equations systematically, matching them to the graphs.### Cubic Equations and Solutions

1. $x^3 - 3x^2 + x + 5 = 0$

From the graph:

• It appears to have one real root.
• The other two solutions are complex conjugates.

2. $x^3 - 2x^2 - x + 2 = 0$

From the graph:

• It shows three distinct real roots crossing the x-axis.

3. $x^3 - x^2 - 4x + 4 = 0$

From the graph:

• There are two real roots (one is a double root as the curve touches the x-axis and turns back).
• The third root is complex.

4. $x^3 - 3x^2 + x - 3 = 0$

From the graph:

• It has one real root.
• The other two solutions are complex.

Conjecture

From the graph of a cubic function:

• Number of real zeros: Each crossing or tangent point of the curve with the x-axis indicates a real root.
• Complex zeros: If the cubic equation has fewer than three x-axis intersections, the missing roots must be complex conjugates.

Would you like me to solve these equations algebraically for verification or clarify any points?

Expanding on the topic:

1. What is the relationship between turning points and real roots?
2. How can synthetic division help in finding one real root?
3. How are the coefficients of the cubic equation related to the nature of its roots?
4. Can Descartes' Rule of Signs help in predicting the number of positive/negative roots?
5. What role does the discriminant of a cubic play in determining complex solutions?

Tip: For any cubic polynomial, always analyze the derivative graph for better root approximation.