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Out of a number of Saras birds, one fourth the number are moving about it lotus plants, ${(\frac{1}{9})}^{th}$ to coupled(along) with ${(\frac{1}{4})}^{th}$ as well as 7times the square root of the number move on a hill, 56 birds remain in vakula trees. Total number of birds will be ____.

Subtopics: The Fundamental Theorem of Arithmetic

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Detailed Solution

Let the number of birds be

x^{2}

Number of birds moving about lotus plant =

\frac{x^{2}}{4}

Number of birds coupled along =

\frac{x^{2}}{9}

Number of bird moves on hill =

7 \sqrt{x^{2}} = 7 x

Number of birds remaining on trees =56=56
Now,

\frac{x^{2}}{4} + \frac{x^{2}}{4} + \frac{x^{2}}{9} + 7 x + 56 = x^{2}

\frac{x^{2} + x^{2}}{4} + \frac{x^{2}}{9} + 7 x + 56 = x^{2}

\frac{{2 x}^{2}}{4} + \frac{x^{2}}{9} + 7 x + 56 - x^{2} = 0

\frac{x^{2}}{2} + \frac{x^{2}}{9} + 7 x + 56 - x^{2} = 0

\frac{9 x^{2} + 2 x^{2}}{18} + 7 x + 56 - x^{2} = 0

\frac{11 x^{2}}{18} - x^{2} + 7 x + 56 = 0

\frac{11 x^{2} - 18 x^{2}}{18} + 7 x + 56 = 0

\frac{- 7 x^{2}}{18} + 7 x + 56 = 0

\frac{- 7 x^{2} + 126 x + 1008}{18} = 0

- 7 x^{2} + 126 x + 1008 = 0 * 18

- 7 {(x}^{2} - 18 x - 144) = 0

x^{2} - 18 x - 144 = 0

x^{2} - 24 x + 6 x - 144 = 0

x (x - 24) + 6 (x - 24) = 0

(x - 24) (x + 6) = 0

(x - 24) = 0 or (x + 6) = 0

x = 24 or   x = - 6

Here we are considering

x = 24

Because number of birds cannot be a negative
The total number of birds

= x^{2} = {(24)}^{2} = 576

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