Question

The sum of the squares of three consecutive natural number is 149. Find the numbers.

Answer 1

$\bf\huge\boxed{\boxed{\boxed{Solution:- }}}$$\huge\mathfrak\red$
Let the Three Consecutive Numbers be x , x+1 and x+2.

A.T.Q.

$= > {x}^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} = 149 \\ \\ = > {x}^{2} + {x}^{2} + 1 + 2x + {x}^{2} + 4 + 4x = 149 \\ \\ = > 3 {x}^{2} + 6x + 5 = 149 \\ \\ = > 3 {x}^{2} + 6x - 144 = 0 \\ \\ = > {x}^{2} + 2x - 48 = 0 \\ \\ = > {x}^{2} + (8 - 6)x - 48 = 0 \\ \\ = > {x}^{2} + 8x - 6x - 48 = 0 \\ \\ = > x(x + 8) - 6(x + 8) = 0 \\ \\ = > (x - 6)(x + 8) = 0 \\ \\ = > x = 6 \: \: \: and \: \: \: x = - 8$

Hence, The Numbers are:-

x = 6
x+1 = 7
x+2 = 8.

Answer 2

SOLUTION :

Let the numbers be x , x + 1, and x + 2  

A.T.Q

x² + (x + 1)² + (x + 2)² = 149

x² + x²  + 1+ 2x+  x² + 4 + 4x = 149

x² + x² + x² + 2x + 4x  + 4  + 1 - 149 = 0

3x²  + 6x -144 = 0

3(x² + 2x - 48 ) = 0

x² + 2x - 48 = 0

By  factorization method

x² + 8x - 6x - 48 = 0

x(x + 8) - 6(x + 8) = 0

(x + 8)(x - 6) = 0

(x + 8) = 0  or (x - 6) = 0

x = - 8 or x = 6  

Since, x is a natural number , so x can't be negative. x ≠ - 8

Therefore , x = 6

First consecutive natural number = x = 6  

Second consecutive natural number = (x + 1) =  6 + 1 = 7  

Third consecutive natural number =( x + 3) = 6 + 2 = 8

Hence, the three consecutive natural numbers are 6 , 7 , and 8 .

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