Question

the graph of $f(x) = x^3 + 6x^2 + 12x + 8$ is shown. based on the graph, how many distinct real number solutions does the equation $x^3 + 6x^2 + 12x + 8 = 0$ have? \bigcirc no real number solutions \bigcirc one real number solution \bigcirc two real number solutions \bigcirc three real number solutions

Explanation:

Step1: Recall the relationship between a function's graph and its equation's solutions

The solutions to the equation \( f(x) = 0 \) are the x - intercepts of the graph of \( y = f(x) \). So we need to find the number of times the graph of \( f(x)=x^{3}+6x^{2}+12x + 8 \) intersects the x - axis.

Step2: Analyze the given graph

Looking at the graph of \( y=x^{3}+6x^{2}+12x + 8 \), we observe that the graph touches the x - axis at \( x=-2 \) (it is a repeated root, but in terms of distinct real number solutions, we count the number of distinct x - values where the graph intersects or touches the x - axis). The graph touches the x - axis at one distinct point (even though it may be a multiple root, it is still one distinct real solution in terms of the number of distinct real x - values that satisfy the equation).

Answer:

B. one real number solution