Question 178136
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Let <i>x</i> be the width of the photo.  Then the length of the photo must be *[tex \Large x + 4]

If the photo has a 3 inch border, then the width is increased by 6 (3 on each side) and the length is also increased by 6, so:

Width of photo plus border *[tex \Large x + 6]

Length of photo plus border *[tex \Large x + 10]

The area of a rectangle is given by *[tex \Large A = lw], so the overall area of the photo and border must be:

*[tex \Large (x + 6)(x + 10) = 165]

Apply FOIL:

*[tex \Large x^2 +16x +160 = 165 \text { } \Rightarrow \text { } \math x^2 +16x -5 = 0]

Use the quadratic formula (or complete the square, if you prefer) to solve this quadratic equation and obtain the width of the photo, <i>x</i>.  The quadratic formula, as you are certainly aware, will provide two solutions.  One of these solutions will be a negative number.  The negative result is an absurdity because we are trying to find the magnitude of the width of something.  This, therefore, is an extraneous root introduced by squaring the variable in the process of solving the problem.  Exclude the negative root and the positive root is your answer.

Since you were not asked for a numerical approximation, I would leave the answer in simplest radical form - that being the exact answer.

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