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The width of a rectangle is the length minus 6 units. the area of the rectangle is 27 units. what is the length in units of the rectangle ?

  9 units

Step-by-step explanation:

If you assume the solution is an integer, then you are looking for factors of 27 that differ by 6. You don't have to look far:

  27 = 1×27 = 3×9

If the length is 9 units and the width is 3 units, the width is equal to the length minus 6 units.

The length is 9 units.

Alternate solutions

You can write an equation for the area using L to represent the length. The area is the product of length and width, so is ...

  A = LW

  27 = L(L-6) = L^2 -6L

You can "complete the square" by adding the square of half the L coefficient:

  27 +3^2 = L^2 -6L +3^2

  36 = (L -3)^2 . . . . . simplify a bit

  6 = L -3 . . . . . . . . . positive square root

  9 = L . . . . . . . . . . . . add 3

The length is 9 units.

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You can also put the quadratic equation into standard form:

  L^2 -6L -27 = 0

To factor this, you're looking for factors of -27 that have a sum of -6. From above, we know these are -9 and +3. So the factorization is ...

  (L -9)(L +3) = 0

The solutions are the values of L that make these factors be zero:

  L -9 = 0   ⇒   L = 9

  L +3 = 0   ⇒   L = -3

Only the positive solution is useful as a measure of length. The length is 9 units.