1. **State the problem:** We need to find the quadratic regression equation of the form $$y = ax^2 + bx + c$$ that best fits the given data points. 2. **Given data:** $$x: 0, 1, 2, 3, 4, 5, 6, 7, 8$$ $$y: 80, 70, 60, 60, 75, 90, 120, 175, 255$$ 3. **Formula and method:** Quadratic regression finds coefficients $a$, $b$, and $c$ that minimize the sum of squared residuals between observed $y$ and predicted $y$. This is typically done using least squares fitting. 4. **Calculate sums needed:** Calculate sums of $x$, $x^2$, $x^3$, $x^4$, $y$, $xy$, and $x^2y$: $$\sum x = 36$$ $$\sum x^2 = 204$$ $$\sum x^3 = 1296$$ $$\sum x^4 = 8652$$ $$\sum y = 885$$ $$\sum xy = 4690$$ $$\sum x^2 y = 31340$$ 5. **Set up normal equations:** $$\begin{cases} 9c + 36b + 204a = 885 \\ 36c + 204b + 1296a = 4690 \\ 204c + 1296b + 8652a = 31340 \end{cases}$$ 6. **Solve the system:** Using matrix methods or substitution, the solution rounded to two decimals is: $$a = 4.39, \quad b = -30.11, \quad c = 81.33$$ 7. **Write the quadratic regression equation:** $$y = 4.39x^2 - 30.11x + 81.33$$ This equation models the data with coefficients rounded to the nearest hundredth.