Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
$\begin{matrix} \begin{matrix} A r e a : 25 a^{2} - 35 a + 12 \end{matrix} (i) \end{matrix} \begin{matrix} \begin{matrix} A r e a : 35 y^{2} + 13 y - 12 \end{matrix} (i i) \end{matrix}$

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Solution

Here, area is given in terms of a quadratic expression in one varaible. To find the possible length and breadth of a rectangle, we have to factorise the given expression.

$(i) A r e a = 25 a^{2} - 35 a + 12 (A r e a = l \times b) = 25 a^{2} - 20 a - 15 a + 12 (s p l i t t i n g m i d d l e t e r m) = 5 a (5 a - 4) - 3 (5 a - 4) = (5 a - 4) (5 a - 3)$
So, possible answer is length = (5a-4), breadth = (5a-3) or vice versa.

$(i i) A r e a = 35 y^{2} + 13 y - 12 (A r e a = l \times b) = 35 y^{2} + 28 y - 15 y - 12 (s p l i t t i n g m i d d l e t e r m) = 7 y (5 y + 4) - 3 (5 y + 4) = (5 y + 4) (7 y - 3)$
So, (5y + 4) may be taken as breadth and (7y - 3) as length.

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Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: Area:25a2−35a+12(i) Area:35y2+13y−12(ii)