1. **State the problem:** Philip takes 9 hours longer than Jane to paint a car alone. Together, they paint the car in 6 hours. We need to find how long Jane takes to paint the car alone. 2. **Devise a plan:** Let $j$ be the time (in hours) Jane takes to paint the car alone. Then Philip takes $j + 9$ hours. The rate of work is the reciprocal of time, so: - Jane's rate: $\frac{1}{j}$ cars per hour - Philip's rate: $\frac{1}{j+9}$ cars per hour Working together, their combined rate is $\frac{1}{6}$ cars per hour. 3. **Carry out the plan:** Set up the equation for combined rates: $$\frac{1}{j} + \frac{1}{j+9} = \frac{1}{6}$$ Multiply both sides by $6j(j+9)$ to clear denominators: $$6(j+9) + 6j = j(j+9)$$ Simplify: $$6j + 54 + 6j = j^2 + 9j$$ $$12j + 54 = j^2 + 9j$$ Bring all terms to one side: $$0 = j^2 + 9j - 12j - 54$$ $$0 = j^2 - 3j - 54$$ Solve the quadratic equation $j^2 - 3j - 54 = 0$ using the quadratic formula: $$j = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2} = \frac{3 \pm \sqrt{9 + 216}}{2} = \frac{3 \pm \sqrt{225}}{2}$$ Calculate the roots: $$j = \frac{3 \pm 15}{2}$$ Two possible solutions: - $j = \frac{3 + 15}{2} = \frac{18}{2} = 9$ - $j = \frac{3 - 15}{2} = \frac{-12}{2} = -6$ Since time cannot be negative, discard $j = -6$. 4. **Look back and reflect:** Jane takes 9 hours to paint the car alone. Philip takes $9 + 9 = 18$ hours. Check combined rate: $$\frac{1}{9} + \frac{1}{18} = \frac{2}{18} + \frac{1}{18} = \frac{3}{18} = \frac{1}{6}$$ This matches the problem statement. **Final answer:** Jane takes $\boxed{9}$ hours to paint the car alone.