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Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(i) $\boxed{Area:\ 25\ a^{2} -35\ a+12}$
(ii) $\boxed{Area\ :\ 35\ y^{2} +13\ y-12\ }$
To do :
We have to find possible expressions for the length and breadth of each of the given rectangles.
Solution :
We know that,
Area of a rectangle of length $l$ and breadth $b$ is $lb$.
Therefore, factorizing the given expressions, we get,
(i) Area $=25a^2 - 35a + 12$
$=25a^2- 20a- 15a + 12$ [Since $25 \times12=300=(-20) \times(-15), -35=-20-15$]
$=5a(5 a-4)-3(5 a-4)$
$= (5 a-3)(5a-4)$
If length $= 5a - 3$, then breadth $= 5a - 4$
If length $= 5a-4$, then breadth $= 5a- 3$
Hence, possible expressions for the length and breadth of the rectangle are $(5 a-4)$ and $(5 a-3)$.
(ii) Area $=35y^2 + 13y - 12$
$=35y^2+ 28y- 15y- 12$ [Since $35 \times(-12)=-420=28 \times(-15), 13=28-15$]
$=7 y(5 y+4)-3(5 y+4)$
$= (5 y+4)(7 y-3)$
If length $= 5y + 4$, then breadth $= 7y - 3$
If length $= 7y-3$, then breadth $= 5y+ 4$
Hence, possible expressions for the length and breadth of the rectangle are $(5 y+4)$ and $(7 y-3)$.
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