First degree or second degree quadratic equation?

Question

$X-\sqrt{121}= 0$

$Y^2-121=0$

When I try to solve by squaring on both sides the first equation changes to $X^2-121 =0$, where $x = \pm 11$.

When I try to solve it this way:

$x= \sqrt{121} = +11$, there is only one solution.

Why the difference?

Answer 1

The equation $x=a$ has just one solution.

$x^2=a^2$ is equivalent to $0=x^2-a^2=(x+a)(x-a)$ with solutions $x=a$ and $x=-a$.

It is a characteristic feature of squaring equations that the squared equation can have solutions which weren't there in the original equation. Any solution derived after squaring should be checked back to the original equation.

Likewise taking the square root (using the convention that the positive real root is taken) can lose solutions of the original equation. This is why the general solution to the quadratic is written with $\pm $ before the square root.