1. **State the problem:** We need to find the length and width of a rectangle where the length is 3 ft less than twice the width, and the area is 27 ft$^2$. 2. **Define variables:** Let the width be $w$ ft. 3. **Express length in terms of width:** Length $L = 2w - 3$ ft. 4. **Write the area formula:** Area $A = L \times w$. 5. **Set up the equation:** Given $A = 27$, so $$ (2w - 3) \times w = 27 $$ 6. **Expand and simplify:** $$ 2w^2 - 3w = 27 $$ 7. **Bring all terms to one side:** $$ 2w^2 - 3w - 27 = 0 $$ 8. **Solve the quadratic equation using the quadratic formula:** The quadratic formula is $$ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=2$, $b=-3$, and $c=-27$. 9. **Calculate the discriminant:** $$ \Delta = (-3)^2 - 4 \times 2 \times (-27) = 9 + 216 = 225 $$ 10. **Calculate the roots:** $$ w = \frac{-(-3) \pm \sqrt{225}}{2 \times 2} = \frac{3 \pm 15}{4} $$ 11. **Find the two possible values for $w$:** - $w = \frac{3 + 15}{4} = \frac{18}{4} = 4.5$ - $w = \frac{3 - 15}{4} = \frac{-12}{4} = -3$ 12. **Interpret the results:** Width cannot be negative, so $w = 4.5$ ft. 13. **Find the length:** $$ L = 2w - 3 = 2 \times 4.5 - 3 = 9 - 3 = 6 \text{ ft} $$ **Final answer:** - Width = 4.5 ft - Length = 6 ft