show that x²+2x=(2a+2b+1)(2a+2b-1) are the root of the equation are integers

Answer 1

The statement, as written, is completely FALSE.

Here is a counter example.

For a=0 and b=sqrt(2),

2*a+2*b + 1 = 2*sqrt(2) + 1

2*a +2*b - 1 = 2*sqrt(2) - 1

Multiplying them: 8 - 1 = 7

Then the quadratic equation becomes:

x^2 + 2x = 7

x^2 - 2x - 7 = 0

Quadratic formula says:

[2 +or- sqrt( 4 - 4*-7) ]/ 2

[2 +or- sqrt( 4- -28)] / 2

[ 2 +or- sqrt(32)] / 2

[2 +or- 4*sqrt(2)]/2

1 +or- 2*sqrt(2)

The solution is IRRATIONAL and not an integer.

Please repost with the correct statement of the problem.