Why Does a^2 + b^2 Have No Solution?

[Anatoly]

Why people say that a^2 + b^2 has no solution?

 

[PeroK]

View attachment 224874
Why people say that a^2 + b^2 has no solution?

Who says that?

 

[mfb]

$a^2+b^2$ is not an equation, “solution” is a meaningless concept.

You can write the expression as product of two factors - so what? The easiest way to do so is writing it as $1(a^2+b^2)$. That doesn’t make it easier.

 

[jedishrfu]

I think the OP is referring to how algebra teachers introduce factoring to say that $a^2 - b^2$ is factorable into $(a -b) * (a + b)$ and that $a^2 + b^2$ is not factorable in the context of real numbers.

https://brownmath.com/alge/sumsqr.htm

 

[Mark44]

View attachment 224874
Why people say that a^2 + b^2 has no solution?

The equation $a^2 + b^2 = 0$ has no solution in the real numbers. If you relax this restriction, allowing complex numbers, then there are solutions.
Edit: $a^2 + b^2 = 0$ does have a solution: a = b = 0. However, the equation $a^2 + b^2 = -1$ has no real solution, as @mfb points out in a subsequent post.

The quadratic (i.e., 2nd degree) polynomial $a^2 + b^2$ cannot be factored into the product of two linear (i.e., first degree) polynomials with real coefficients. Your factorization, with terms involving $\sqrt{2ab}$ doesn't consist of linear polynomials. In fact, $a \pm \sqrt{2ab} + b$ isn't even a polynomial of any kind.

As already mentioned, equations and inequalities have solutions, but expressions don't.

 

[mfb]

The equation $a^2 + b^2 = 0$ has no solution in the real numbers. If you relax this restriction, allowing complex numbers, then there are solutions.

$a=b=0$ is a solution.
$a^2 + b^2 = -1$ has no real solution.

 

[Mark44]

$a=b=0$ is a solution.

Well, yes, there's that one. Doh!