Question

the length of a rectangle is 8 more than the width. the area is 273 square inches. find the length and width of the rectangle

Answer 1

273 = 1·273 = 3·91 = 7·39 = 13·21

The factors that differ by 8 are 13 and 21.

The length is 21 inches; the width is 13 inches.


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If you're going to write an equation for width (w), you can write it as
  w(w+8) = 273
and simplify it to
  w² +8w -273 = 0
Solving this by factoring requires you find the factors of 273 that differ by 8, as above.

You can also solve by completing the square or using the quadratic formula. Those methods will tell you w is either of -21 or 13. Only the positive solution is applicable.

Another method I use mentally is to think of the equation in terms of the average of length and width (x). Then the area is
  (x-4)(x+4) = 273
  x² -16 = 273 . . . . . this is a much easier quadratic to solve
  x² = 289
  x = 17 . . . . . . . . you know this because you have the squares of small numbers memorized.
Then the dimensions are 17-4 = 13, and 17+4 = 21.