You have made a rectangular stained glass window that is $2$ feet by $4$ feet. You have 7 square feet of clear glass to create a border of uniform width around the window. What should the width of the border be?
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1 Answer
$\begingroup$The total area of the window, including the border and the stained glass, is the product of the height and width: $h \cdot w = (4 + 2x)(2 + 2x) = 4x^2 + 12x + 8\ \text{ft}^2$. The area of the stained glass is $2 \cdot 4 = 8\ \text{ft}^2$, so the area of the border is $4x^2 + 12x + 8 - 8\ \text{ft}^2$, which is given to be $7\ \text{ft}^2$. Thus $$4x^2 + 12x - 7 = 0 \\ \implies 4x^2 -2x + 14x - 7 = 0 \\ \implies 2x(2x - 1) + 7(2x - 1) = 0 \\ \implies (2x - 1)(2x + 7) = 0 \\ \implies x = \frac{1}{2}, \frac{-7}{2} $$
Obviously, the width of the border can't be negative, so $x = \frac{1}{2}\ \text{ft}^2$.
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