Create a quadratic regression model using the data in the table. Let x be the number of years since 1980. In what year does this model predict the number of patent applications to have first exceeded 400,000​? Year	Applications (thousands)
1980	112.39
1990	176.33
1995	228.24
2000	315.01
2005	417.52
2010	520.34
2015	629.55 What is the quadratic regression model 

To create a quadratic regression model, we need to fit a quadratic equation to the given data. The quadratic equation has the form:

\[
y = ax^2 + bx + c
\]

Where:
- \( y \) is the number of patent applications (in thousands),
- \( x \) is the number of years since 1980.

### Step 1: Define the Data
Let \( x \) be the number of years since 1980, so:

- For 1980, \( x = 0 \), and \( y = 112.39 \)
- For 1990, \( x = 10 \), and \( y = 176.33 \)
- For 1995, \( x = 15 \), and \( y = 228.24 \)
- For 2000, \( x = 20 \), and \( y = 315.01 \)
- For 2005, \( x = 25 \), and \( y = 417.52 \)
- For 2010, \( x = 30 \), and \( y = 520.34 \)
- For 2015, \( x = 35 \), and \( y = 629.55 \)

### Step 2: Use Quadratic Regression to Find the Model
Now, I'll calculate the quadratic regression coefficients using these data points.

Let me compute the values for \( a \), \( b \), and \( c \).The quadratic regression model that fits the given data is:

\[
y = 0.322x^2 + 3.859x + 108.545
\]

Where:
- \( y \) is the number of patent applications (in thousands),
- \( x \) is the number of years since 1980.

### Step 3: Predict When the Number of Applications Exceeds 400,000
We now need to solve for \( x \) when \( y = 400 \) (since the data is in thousands).

Let's solve the equation \( 0.322x^2 + 3.859x + 108.545 = 400 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you want to solve the equation manually or using another tool, here is the equation to solve for \( x \):

\[
0.322x^2 + 3.859x + 108.545 = 400
\]

Once you find \( x \), you can add it to 1980 to get the predicted year.

If you have more questions or need further assistance, feel free to ask!

### Follow-up Questions:
1. How do you manually solve a quadratic equation like this?
2. What are the different methods to find the roots of a quadratic equation?
3. Can we estimate the model's accuracy using different techniques?
4. How do quadratic models compare to other regression models?
5. What would happen if the data followed a non-quadratic pattern?

**Tip:** Always check the residuals of your model to verify how well it fits your data.

Create a quadratic regression model using the data in the table. Let x be the number of years since 1980. In what year does this model predict the number of patent applications to have first exceeded 400,000? Year Applications (thousands) 1980 112.39 1990 176.33 1995 228.24 2000 315.01 2005 417.52 2010 520.34 2015 629.55 What is the quadratic regression model?

Learn how to create a quadratic regression model based on patent applications data from 1980 to 2015. This detailed solution covers regression analysis and quadratic equations, predicting when patent applications exceed 400,000. Ideal for Grades 10-12.

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