Select all the points of intersection between the graphs of the functions  f(x)=(x+5)(x−2)
  and  g(x)=(2x+1)(x−2).
 

Multiple select question.
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A)
(−5,0)
 

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B)
(−12,0)
 

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C)
(−2,−12)
 

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D)
(2,0)
 

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E)
(4,18)
 

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F)
(5,30)

To find the points of intersection between the graphs of the functions \( f(x) = (x+5)(x-2) \) and \( g(x) = (2x+1)(x-2) \), we need to set the two functions equal to each other and solve for \( x \).

### Step 1: Set the equations equal to each other.
\[
(x+5)(x-2) = (2x+1)(x-2)
\]

Since both sides contain \( (x-2) \), we can divide both sides of the equation by \( (x-2) \) as long as \( x \neq 2 \).

\[
x+5 = 2x+1
\]

### Step 2: Solve for \( x \).
\[
x+5 = 2x+1
\]
Subtract \( x \) from both sides:
\[
5 = x+1
\]
Subtract 1 from both sides:
\[
x = 4
\]

So, one point of intersection occurs when \( x = 4 \).

### Step 3: Find the corresponding \( y \)-coordinate.
Substitute \( x = 4 \) into either function to find the \( y \)-value. Let's use \( f(x) \):
\[
f(4) = (4+5)(4-2) = 9 \times 2 = 18
\]
Thus, the point of intersection is \( (4, 18) \).

### Step 4: Check if \( x = 2 \) is also a solution.
Since both functions have \( (x-2) \) as a factor, let's substitute \( x = 2 \) into both functions:
\[
f(2) = (2+5)(2-2) = 7 \times 0 = 0
\]
\[
g(2) = (2 \times 2 + 1)(2-2) = 5 \times 0 = 0
\]
Both functions give \( y = 0 \) when \( x = 2 \). Therefore, \( (2, 0) \) is also a point of intersection.

### Step 5: Conclusion
The points of intersection are \( (4, 18) \) and \( (2, 0) \).

The correct selections are:
- **D)** \( (2, 0) \)
- **E)** \( (4, 18) \)

Would you like more details on the calculations or further explanation?

Here are 5 related questions to explore:

1. How do you factor quadratic equations like \( f(x) = (x+5)(x-2) \)?
2. How would the points of intersection change if the functions were modified?
3. What is the significance of dividing by \( (x-2) \) in this problem?
4. Can there be cases where dividing by a common factor might lose valid solutions?
5. How can you graphically verify the points of intersection?

**Tip:** Always check if common factors can affect the domain of your solution before dividing through them!

Select all the points of intersection between the graphs of the functions f(x) = (x+5)(x−2) and g(x) = (2x+1)(x−2).

Discover how to find the points of intersection between the quadratic functions f(x) = (x+5)(x-2) and g(x) = (2x+1)(x-2) using factoring and solving techniques. This step-by-step guide is ideal for students in Grades 9-10.

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