1. **State the problem:** A photograph originally measures 16 cm by 11 cm. It is cropped by removing the same width $x$ cm from the top and left side. The area is reduced by 72 cm². We need to find the dimensions of the cropped photograph. 2. **Set up the equation:** Original area = $16 \times 11 = 176$ cm². After cropping, the new dimensions are: - Width: $16 - x$ - Height: $11 - x$ New area = $(16 - x)(11 - x)$. 3. **Express the area reduction:** The area is reduced by 72 cm², so: $$176 - (16 - x)(11 - x) = 72$$ 4. **Simplify the equation:** Expand the product: $$(16 - x)(11 - x) = 16 \times 11 - 16x - 11x + x^2 = 176 - 27x + x^2$$ Substitute back: $$176 - (176 - 27x + x^2) = 72$$ Simplify: $$176 - 176 + 27x - x^2 = 72$$ $$27x - x^2 = 72$$ Rewrite as: $$-x^2 + 27x - 72 = 0$$ Multiply both sides by -1: $$x^2 - 27x + 72 = 0$$ 5. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{27 \pm \sqrt{27^2 - 4 \times 1 \times 72}}{2}$$ Calculate the discriminant: $$27^2 - 4 \times 72 = 729 - 288 = 441$$ Square root: $$\sqrt{441} = 21$$ So: $$x = \frac{27 \pm 21}{2}$$ Two possible solutions: - $$x = \frac{27 + 21}{2} = \frac{48}{2} = 24$$ - $$x = \frac{27 - 21}{2} = \frac{6}{2} = 3$$ 6. **Check for valid solution:** Since $x$ is the width removed, it must be less than the original dimensions (16 and 11 cm). $x=24$ is invalid because it exceeds the dimensions. Therefore, $x = 3$ cm. 7. **Find the cropped dimensions:** Width: $16 - 3 = 13$ cm Height: $11 - 3 = 8$ cm **Final answer:** The cropped photograph measures 13 cm by 8 cm.