TargetIIT

[b](1) find all  Integral value of x for which the expression
[im]http://codecogs.izyba.com/gif.latex?x%5E%7B2%7D+7x+13[/im]
is a perfect square.

i have got only one solution x= - 4

is there is any other Integral value of x which satisfy this expression?[/b]

[im]http://codecogs.izyba.com/gif.latex?x%5E2+7x+13[/im] is a square [im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%204x%5E2+28x+52%20%3D%20%282x+7%29%5E2+3[/im] is a perfect square.

If m[p]2[/p], n[p]2[/p] are both distinct greater than 4, their difference is greater than 4. Hence

we must have [im]http://codecogs.izyba.com/gif.latex?%7C2x+7%7C%3D1[/im] which has solutions -3,-4

[b]sir i have a problem in understanding (2) and (3) line.
[/b]

understand it this way
[im]http://codecogs.izyba.com/gif.latex?3%3Dk%5E2-%282x+7%29%5E2%20%5C%5C%20%5CRightarrow%203%3D%5Cleft%28k-%282x+7%29%20%5Cright%29%5Cleft%28k+%5Cleft%282x+7%20%5Cright%29%20%5Cright%29%5C%5C%20since%20%5C%203%20%5C%20is%20%5C%20prime%20%5C%20%5C%5C%20%5Cbegin%20%7Bcases%7D%20k-%5Cleft%282x+7%20%5Cright%29%3D3%20%5C%5C%20k+%5Cleft%282x+7%20%5Cright%29%3D1%5Cend%7Bcases%7D%20%5C%5C%20%5Cbegin%20%7Bcases%7D%20k-%5Cleft%282x+7%20%5Cright%29%3D1%20%5C%5C%20k+%5Cleft%282x+7%20%5Cright%29%3D3%5Cend%7Bcases%7D%20%5C%5C%20%5Cbegin%20%7Bcases%7D%20k-%5Cleft%282x+7%20%5Cright%29%3D-3%20%5C%5C%20k+%5Cleft%282x+7%20%5Cright%29%3D-1%5Cend%7Bcases%7D%20%5C%5C%20%5Cbegin%20%7Bcases%7D%20k-%5Cleft%282x+7%20%5Cright%29%3D-1%20%5C%5C%20k+%5Cleft%282x+7%20%5Cright%29%3D-3%5Cend%7Bcases%7D%20%5Ctexttt%7Bsolve%20these%20equations%20to%20get%20x%27s%7D[/im]

no that's not what I had in mind.

We are claiming that (2x+7)[p]2[/p]+3 is a perfect square.

But (2x+7)[p]2[/p] is itself a perfect square.

If you look at the sequence of squares 0,1,4,9,16,25,... the difference between them even consecutive squares increasing.

Hence, the only squares whose difference is 3 are 1 and 4.

So |2x+7|=1

[b]Thanks hsbhatt sir.
and also thanks to xYz.[/b]