f(x)=x^2+3x+5

g(x)=5


Solve ​f(x)equals​g(x). What are the points of intersection of the graphs of the two​ functions?

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!

Solve f(x) = x^2 + 3x + 5 and g(x) = 5. What are the points of intersection of the graphs of the two functions?

Find the points of intersection of the functions f(x) = x^2 + 3x + 5 and g(x) = 5 with a step-by-step solution. This problem covers algebraic concepts like quadratic equations, factoring, and graphing. Perfect for students in Grades 8-10.

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