A rectangular garden is to be designed such that its length is $5$ m more than its breadth. The area of the garden must be $84{\text{ m}}^2$. Find the dimensions of the garden.

Step-by-step solution

1. Let breadth be $x$ metres; then length = $x+5$ metres (given).
2. Write the area condition: $x(x+5)=84$.
3. Expand: $x^2+5x-84=0$.
4. Factor the quadratic: numbers whose product is $-84$ and sum $5$ are $12$ and $-7$.
5. Write as $(x+12)(x-7)=0$.
6. Set each factor to zero: $x+12=0$ gives $x=-12$ (reject negative breadth); $x-7=0$ gives $x=7$.
7. Hence breadth $=7$ m and length $=7+5=12$ m.

Concept tested: Word problem – forming quadratic

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