Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: Area: `25 a^2-35 a+12` Area: `35 y^2+13 y-12`

Explanation

Solution:
i) Area of rectangle = `25a² - 35a + 12`
But we know that, Area of rectangle =length x breadth
Hence, we shall factorise the given expression `25a² - 35a + 12`
Now taking `25a² - 35a + 12`,
find two numbers p, q such that:
`p+q`=co-efficient of a
• pq = product of the co-efficient of `a²` and the constant
`p+q=-35 (co-efficient of a)
pq = 25 x 12 = 300` (product of the co-efficient of a² and the constant term.)
By trial and error method, we get `p = -20, q = -15`.
Now splitting the middle term of the given polynomial,
`25a²-35a +12= 25a2 - 20a - 15a + 12
= 25a² - 15a - 20a + 12
= 5a(5a - 3) - 4(5a - 3)
= (5a-4) (5a - 3)
25a²-35a + 12 = (5a - 4) (5a - 3)`
Thus, Length `5a - 3, Breadth = 5a - 4(OR) Length 5a - 4, Breadth = 5a - 3
ii) Area of rectangle = 35y² + 13y - 12`
But we know that, `Area of rectangle =length x breadth`
Hence, we shall factorise the given expression `35y² + 13y - 12`Now taking `35y² + 13y - 12`, find two numbers p, q such that:
`p+q`=co-efficient of y
pq = product of the co-efficient of y² and the constant
`p+q=13 (co-efficient of y)
pq = 35 × (- 12) = - 420 (product of the co-efficient of y² and the constantterm.)`
By trial and error method, we get `p = 28, q = -15.`
Now splitting the middle term of the given polynomial,
`35y² + 13y - 12`
= `35y²+28y-15y - 12
= 7y(5y + 4) - 3(5y + 4)
= (5y + 4) (7y - 3)
:: 35y²+13y - 12 (5y + 4) (7y -3)
Hence, Length = 5y + 4, Breadth = 7y - 3
(OR) Length = 7y - 3, Breadth = 5y + 4`