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.