1. **State the problem:** A boy is three years older than his sister. The product of their ages is 54. We need to find their ages. 2. **Define variables:** Let the sister's age be $x$ years. 3. **Express the boy's age:** Since the boy is three years older, his age is $x + 3$. 4. **Write the equation for the product of their ages:** $$x(x + 3) = 54$$ 5. **Expand the equation:** $$x^2 + 3x = 54$$ 6. **Bring all terms to one side to form a quadratic equation:** $$x^2 + 3x - 54 = 0$$ 7. **Solve the quadratic equation using the quadratic formula:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=3$, and $c=-54$. 8. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4(1)(-54) = 9 + 216 = 225$$ 9. **Find the square root of the discriminant:** $$\sqrt{225} = 15$$ 10. **Calculate the two possible values for $x$:** $$x = \frac{-3 \pm 15}{2}$$ 11. **Evaluate each case:** - Case 1: $$x = \frac{-3 + 15}{2} = \frac{12}{2} = 6$$ - Case 2: $$x = \frac{-3 - 15}{2} = \frac{-18}{2} = -9$$ 12. **Interpret the results:** Age cannot be negative, so $x = 6$. 13. **Find the boy's age:** $$6 + 3 = 9$$ 14. **Final answer:** The sister is 6 years old and the boy is 9 years old.