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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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