Question

(9) find the real roots of $y = x^3 - 3x^2 + x - 3$ by inspecting the graph:

Explanation:

Step1: Define real roots condition

Real roots occur where $y=0$, i.e., the graph intersects the $x$-axis ($y=0$ line).

Step2: Inspect graph for intersections

Locate points on the given graph of $y=x^3-3x^2+x-3$ that cross $y=0$. The graph intersects the $x$-axis at $x=3$.

Step3: Verify the root

Substitute $x=3$ into the equation:
$$3^3 - 3(3)^2 + 3 - 3 = 27 - 27 + 3 - 3 = 0$$
This confirms $x=3$ is a root. The other roots are complex, so only $x=3$ is real.

Answer:

3