1. **State the problem:** We are given a rectangle with area 36 cm². The length is labeled as $x$ and the width as $x - 5$. We need to find the dimensions of the rectangle. 2. **Formula used:** The area $A$ of a rectangle is given by the formula: $$A = \text{length} \times \text{width}$$ 3. **Set up the equation:** Substitute the given expressions for length and width: $$x \times (x - 5) = 36$$ 4. **Expand and simplify:** $$x^2 - 5x = 36$$ 5. **Bring all terms to one side to form a quadratic equation:** $$x^2 - 5x - 36 = 0$$ 6. **Solve the quadratic equation using the quadratic formula:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, and $c=-36$. 7. **Calculate the discriminant:** $$b^2 - 4ac = (-5)^2 - 4(1)(-36) = 25 + 144 = 169$$ 8. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{169}}{2(1)} = \frac{5 \pm 13}{2}$$ 9. **Calculate each root:** - $$x = \frac{5 + 13}{2} = \frac{18}{2} = 9$$ - $$x = \frac{5 - 13}{2} = \frac{-8}{2} = -4$$ 10. **Interpret the results:** Since length cannot be negative, discard $x = -4$. 11. **Find the width:** $$x - 5 = 9 - 5 = 4$$ 12. **Final answer:** The dimensions of the rectangle are length = 9 cm and width = 4 cm.