The sum of a number and its positive square root is 6 25 . Find the number.

Let the number be x . By the given data, x + x = 6 25 Let x = y 2 Then, y 2 + y = 6 25 25 y 2 + 30 y - 5 y - 6 = 0 5 y ( 5 y + 6 ) - 1 ( 5 y + 6 ) = 0 ( 5 y + 6 ) ( 5 y - 1 ) = 0 5 y - 1 = 0 ⇒ y = 1 5 ( 5 y + 6 ) = 0 ⇒ y = - 6 5 Since y cannot be a negative number as it is under square root sign. So, y = 1 5 ⇒ x = y 2 = 1 5 2 = 1 25 Thus, value of x is 1 25 .

The sum of a number and its square is 63 4 , find the numbers.

Let the number be x , then its square is x 2 . By the given data, we can write, x + x 2 = 63 4 ⇒ 4 ( x + x 2 ) = 63 ⇒ 4 x 2 + 4 x - 63 = 0 Factorising the equation by splitting the middle term, we get, ⇒ 4 x 2 + 18 x - 14 x - 63 = 0 ⇒ 2 x ( 2 x + 9 ) - 7 ( 2 x + 9 ) = 0 ⇒ ( 2 x + 9 ) ( 2 x - 7 ) = 0 ⇒ ( 2 x + 9 ) = 0 ⇒ x = - 9 2 Also, ⇒ ( 2 x - 7 ) = 0 ⇒ x = 7 2 Thus, the value of x is - 9 2 or 7 2 .