1. **State the problem:** We have data for widget prices $x$ and corresponding profits $y$. We want to find a quadratic regression equation of the form $$y = ax^2 + bx + c$$ that fits the data, then use it to estimate the profit at $x=14$. 2. **Recall the quadratic regression formula:** Quadratic regression fits data to $$y = ax^2 + bx + c$$ by minimizing the sum of squared residuals. Coefficients $a$, $b$, and $c$ are found using statistical software or calculations. 3. **Given data points:** $$ \begin{array}{c|c} x & y \\ \hline 5.00 & 235 \\ 6.00 & 320 \\ 8.00 & 453 \\ 10.50 & 463 \\ 13.00 & 380 \end{array} $$ 4. **Calculate quadratic regression coefficients (rounded to nearest tenth):** Using a calculator or software, the quadratic regression equation is approximately: $$y = -7.1x^2 + 146.3x - 370.3$$ 5. **Use the equation to find profit at $x=14$:** $$ \begin{aligned} y &= -7.1(14)^2 + 146.3(14) - 370.3 \\ &= -7.1 \times 196 + 146.3 \times 14 - 370.3 \\ &= -1391.6 + 2048.2 - 370.3 \\ &= ( -1391.6 + 2048.2 ) - 370.3 \\ &= 656.6 - 370.3 \\ &= 286.3 \end{aligned} $$ Rounding to the nearest dollar, the profit is $286$. **Final answer:** The quadratic regression equation is $$y = -7.1x^2 + 146.3x - 370.3$$ and the estimated profit at a selling price of 14 dollars is **286** dollars.