Question

The sum of a number and its positive square root is $\frac{6}{25}$. Find the number.

Answer 1

SOLUTION :  

Let the number be x and  square root of a number be √x .

A.T.Q  

x +  √x = 6/25

Let √x =  a

a² + a = 6/25

25(a² + a) = 6

25a² + 25a - 6 = 0

25a² - 5a + 30a - 6 = 0

[By middle term splitting method]  

5a(5a - 1) + 6(5a - 1) = 0

(5a + 6) (5a - 1) = 0

(5a + 6) = 0  or  (5a - 1) = 0

5a = - 6  or  5a = 1  

a = - 6/5 or  a = 1/5

since, the number is positive, so a ≠ - 6/5 . Therefore , a = ⅕

√x =  a

√x = 1/5

x = (1/5)²

x = 1/25

Hence, the Required number be 1/25.

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Answer 2

Solution :

Let the number = x

It's positive square root = √x

According to the problem given ,

x + √x = 6/25

=> 25( x + √x ) = 6

=> 25x + 25√x - 6 = 0

=> 25( √x )² + 25√x - 6 = 0

Splitting the middle term , we get

=> 25( √x )² + 30√x - 5√x - 6 = 0

=> 5√x( 5√x + 6 ) - 1( 5√x + 6 ) = 0

=> ( 5√x + 6 )( 5√x - 1 ) = 0

=> 5√x + 6 = 0 or 5√x - 1 = 0

=> √x = -6/5 or √x = 1/5

=> x = ( -6/5 )² or x = ( 1/5 )²

=> x = 36/25 or x = 1/25

Therefore ,

Required number = x = 36/25 or x = 1/25

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