Math CentralQuandaries & Queries


Question from Lucas, a student:

The area of a rectangle is given by A=x2+18x+72
-use factoring to find an expression for the dimensions of the rectangle.
-if the area of the rectangle is 7 square feet, what are the possible values of x?
-what are the dimensions of a rectangle?

Hi Lucas,

You don't know what $x$ represents in this problem but you do know that there is a rectangle which has an area given by $x^2 + 18x + 72$ square feet. However the area of a rectangle is the length times the width so if you can factor $x^2 + 18x + 72$ into two factors then one is the length and the other is the width.

You are then told that the area of the rectangle is actually 7 square feet so

\[x^2 + 18 x + 72 = 7.\]

Rewrite this equation in the form $x^2 + ax + b = 0$ and again you should be able to factor the left side and hence solve for $x.$ This will give you two values for $x.$ Substitute these values for $x$ into the factored expression for the area to obtain the dimensions of two possible rectangles.

The fact that there are two solutions means that you might have two rectangles, one rectangle and another with dimensions that don't make sense or two situations that don't give rectangles.

I hope this helps,

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