Question

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

Answer 1

$25 {x}^{2} - 35x + 12 \\ x = \frac{ - b \frac{ + }{} \sqrt{ {b}^{2} - 4ac } }{2a} \\ x = \frac{35 \frac{ + }{} \sqrt{ {35}^{2} - 4 \times 12 \times 25} }{50} \\ x= \frac{35 \frac{ + }{} \sqrt{1225 - 1200} }{50} = \frac{35 \frac{ + }{} \sqrt{25} }{50} \\ \\ x = \frac{35 + 5}{50} = \frac{40}{50} = \frac{4}{5} \\ x = \frac{35 - 5}{50} = \frac{30}{50} = \frac{3}{5}$

Since 4/5 and 3/5 are the zeroes,

x = 3/5 and x = 4/5 are the roots

5x-3 = 0 and 5x-4 = 0 (Cross Multiply)

Therefore, 25x²-35x+12 = (5x-3)(5x-4)

hence the possible length and breadth of the rectangle are 5x-3 and 5x-4

$35 {y}^{2} + 13y - 12 \\y = \frac{ - b \frac{ + }{} \sqrt{ {b}^{2} - 4ac } }{2a} \\ y = \frac{ - 13 \frac{ + }{} \sqrt{ {13}^{2} - 4 \times 35 \times - 12 } }{2 \times 35} \\ y = \frac{ - 13 \frac{ + }{} \sqrt{ 169 + 1680 } }{70} \\ y = \frac{ - 13 \frac{ + }{} \sqrt{1849} }{70} = \frac{ - 13 \frac{ + }{}43 }{70} \\ \\ y = \frac{ - 13 + 43}{70} = \frac{30}{70} = \frac{3}{7} \\ y = \frac{ - 13 - 43}{70} = \frac{ - 56}{70} = \frac{ - 8}{10} = \frac{ - 4}{5}$

Since 3/7 and -4/5 are the zeroes

y = 3/7 and y = -4/5 are the roots

7y-3 = 0 and 5y+4 = 0 (Cross Multiply)

Therefore, 35y²+13y-12 = (7y-3)(5y+4)

hence the possible length and breadth of the rectangle are 7y-3 and 5y+4

Hope this is helpful to you

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