Quick answer: In a rectangle problem, write area = length × breadth, express both dimensions using one variable, then solve the resulting quadratic. Reject roots that make a length nonpositive.
Worked example
A rectangle is 3 cm longer than it is wide and has area 40 cm². Find its dimensions.
- Let width be x cm, so length is x + 3 cm.
- x(x + 3) = 40 gives x² + 3x − 40 = 0.
- Factor: (x + 8)(x − 5) = 0.
- x = 5 or −8. Only x = 5 is a possible width, giving length 8 cm.
Check the result
5 × 8 = 40 cm² and 8 − 5 = 3 cm.
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Common mistake to avoid
The negative root can be algebraically correct but physically invalid. State why it is rejected rather than silently dropping it.
Try it yourself
1. Area is 54 cm² and length exceeds width by 3 cm. 2. A square has area 49 cm²; find its side.
Answers and reasoning
1. x² + 3x − 54 = (x + 9)(x − 6), so dimensions are 6 cm and 9 cm. 2. Positive side 7 cm, not −7 cm.
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