1. **State the problem:** We need to find the length and width of a rectangular room where the length is 5 yards more than the width, and the area is 150 square yards. 2. **Define variables:** Let the width be $w$ yards. 3. **Express length in terms of width:** Length $l = w + 5$ 4. **Use the area formula for a rectangle:** Area $A = l \times w$ 5. **Substitute the known values:** $150 = (w + 5) \times w$ 6. **Write the equation:** $w(w + 5) = 150$ 7. **Expand the equation:** $w^2 + 5w = 150$ 8. **Bring all terms to one side:** $w^2 + 5w - 150 = 0$ 9. **Solve the quadratic equation using the quadratic formula:** $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=5$, and $c=-150$. 10. **Calculate the discriminant:** $$\Delta = 5^2 - 4 \times 1 \times (-150) = 25 + 600 = 625$$ 11. **Calculate the roots:** $$w = \frac{-5 \pm \sqrt{625}}{2} = \frac{-5 \pm 25}{2}$$ 12. **Find the two possible values for $w$:** - $w = \frac{-5 + 25}{2} = \frac{20}{2} = 10$ - $w = \frac{-5 - 25}{2} = \frac{-30}{2} = -15$ (not possible since width cannot be negative) 13. **Select the valid width:** $w = 10$ yards 14. **Find the length:** $$l = w + 5 = 10 + 5 = 15$$ yards **Final answer:** Length = 15 yards, Width = 10 yards