Quadratic Word Problem: Find the Rectangle's Dimensions

A classic algebra problem, worked out step by step by Dr. E — with the full reasoning behind every move, a video walkthrough, and a free practice worksheet.

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The Problem
Area = 48 length = w + 2 w
Figure not drawn to scale

The area of a rectangle is 48 square units. Its length is 2 units more than its width. Find the dimensions of the rectangle.

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Video solution coming soon. Dr. E walks through this exact problem on the whiteboard, step by step.

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Step-by-Step Solution
1
Name the unknown.
Let the width be w. The length is 2 more, so the length is w + 2. Choosing the smaller side as the variable keeps the algebra clean.
2
Translate “area = 48” into an equation.
Area of a rectangle is length × width: w(w + 2) = 48.
3
Expand and set equal to zero.
w² + 2w − 48 = 0.
Getting one side to zero is what lets us factor.
4
Factor and solve.
Two numbers that multiply to −48 and add to +2 are +8 and −6: (w + 8)(w − 6) = 0, so w = −8 or w = 6. A width can't be negative, so reject −8.
5
Find both dimensions.
Width = 6, length = 6 + 2 = 8.
Check: 6 × 8 = 48. ✓
Answer width = 6, length = 8 The key move is turning the sentence into w(w + 2) = 48, then bringing everything to one side so the quadratic factors.

💡 Why this works — the idea to remember

Almost every "find the dimensions" problem follows three beats: name one side, write the other in terms of it, multiply for the area. That always gives a quadratic, and the negative solution is the one you discard because lengths are positive.

Extension question: if the area were 50, the same setup gives w² + 2w − 50 = 0, which doesn't factor nicely — a perfect lead-in to the quadratic formula.

📄 Practice this skill — free worksheets Printable algebra worksheets with answer keys — more problems just like this one. ➗ All algebra video lessons From foundations to quadratics — free, in English and Filipino. 🎬 Numberbender on YouTube 700+ free math lessons by Dr. E. Subscribe for new solved problems.
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