To solve for the points of intersection of the graphs of $f(x) = x^2 + 3x + 5$ and $g(x) = 5$, we need to set $f(x)$ equal to $g(x)$ and solve for $x$:

$f(x) = g(x)$

$x^2 + 3x + 5 = 5$

Step 1: Simplify the equation

Subtract 5 from both sides:

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

$x^2 + 3x = 0$

Step 2: Factor the equation

Factor out $x$:

$x(x + 3) = 0$

Step 3: Solve for $x$

Set each factor equal to zero:

1. $x = 0$
2. $x + 3 = 0$ → $x = -3$

Step 4: Find the corresponding $y$-values

Since $g(x) = 5$ for both values of $x$, the corresponding $y$-value is 5.

Thus, the points of intersection are:

• $(0, 5)$
• $(-3, 5)$

Final Answer:

The points of intersection of the graphs are $(0, 5)$ and $(-3, 5)$.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the solution change if $g(x)$ were $6$ instead of $5$?
2. How do the graphs of $f(x)$ and $g(x)$ look on a coordinate plane?
3. What is the general method for solving systems of equations involving functions?
4. How do we interpret the intersection points in terms of the graphs of $f(x)$ and $g(x)$?
5. What are some applications of finding intersection points of two functions?

Tip: When solving quadratic equations, always check if they can be factored before using the quadratic formula—it saves time!