Question

Out of a number of birds ,one-fourth of the number are moving about lotus plants 1 upon 9th along with 1 upon 4th as well as 7 times the square root of the number movesbon a hill 56 birds remain in vakula trees .what is the total number of birds
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Answer 1

Answer:

$\Large{\underline{\underline{\bf{QuEsTiOn:-}}}}$

Out of a number of birds ,one-fourth of the number are moving about lotus plants 1 upon 9th along with 1 upon 4th as well as 7 times the square root of the number movesbon a hill 56 birds remain in vakula trees .what is the total number of birds .

$\huge\underline{\overline{\mid{\bold{\pink{ANSWER-}}\mid}}}$

Let us Consider that number of Birds are = $\sf x^2$

The number of Birds moving about lotus plants = $\sf\dfrac{x^2}{4}$

Number of Birds along = $\sf\dfrac{x^2}{9} + \dfrac{x^2}{4}$

And, the Number of Birds who are moving on a Hill = $\sf \: 7 \sqrt{x^2}$

$:\implies\sf 7x$

Remain Birds = 56

As Per Given Question -

$:\implies\sf \dfrac{x^2}{4} + \dfrac{x^2}{4} + \dfrac{x^2}{9} + 7x + 56 = x^2$

$:\implies\sf \dfrac{11 \: x^2}{18} + 7x + 56 = x^2$

$:\implies\sf 7x + 56 = \dfrac{7 x^2}{18}$

$:\implies\sf 7x^2 - 136x - 1008 = 0$

$:\implies\sf x^2 - 18x - 144 = 0$

$:\implies\sf x^2 - 24x + 6x - 144 = 0$

$:\implies\sf x(x - 24) + 6(x - 24) = 0$

Now, Comparing these factors with 0.

$:\implies\sf x - 24 = 0$

$:\implies\sf\red{x = 24}$

$:\implies\sf x + 6 = 0$

$:\implies\sf\red{x = -6}$

__________________

$:\implies\sf x^2 \:\:\:\:\:\:\:\:\:\:\:\:\:\:\ [ x = 24]$

$:\implies\sf 24 \times 24$

$:\implies\sf\pink{576}$

$\mathfrak{\huge{\blue{\underline{\underline{Hemce :}}}}}$

Total Number of Birds are = 576.

Answer 2

Let the total number of birds = x

Birds moving in lotus plants = $\sf{\frac{1}{4}x}$

Birds moving on a hill = $\sf{\frac{1}{9} x + \frac{1}{4} x + 7 \sqrt{x} }$

According to the question, the total number of birds will be

= $\sf{\frac{1}{4} x + \frac{1}{9} x + \frac{1}{4} x + 7 \sqrt{x} + 56 = x}$

Multiplying each term by 36, we get

= $\sf{9x + 4x + 9x - 36x + 7 \times 36 \sqrt{x} + 56 \times 36 = 0}$

= $\sf{- 14x + 7 \times 36 \sqrt{x} + 56 \times 36 = 0}$

Dividing both sides by (-14), we get

= $\sf{x-18√x-144=0}$

Is is Quadratic in √x

= $\sf{ {( \sqrt{x} )}^{2} - 24 \sqrt{x} + 6 \sqrt{x} - 144 = 0}$

= $\sf{ \sqrt{x} ( \sqrt{x} - 24) + 6( \sqrt{x} - 24) = 0}$

= $\sf{( \sqrt{x} + 6)( \sqrt{x - 24)} = 0}$

= $\sf{\sqrt{x} = 24 \: or \: \sqrt{x} = - 6}$

= $\sf{x = {(24)}^{2} = 576}$

Hence, the total number of birds = 576