Answered: "Domain of x function when solving x2−2=0"

High School
Algebra
Algebra I
Quadratic function and equation

Amya Horn

Answered question

2022-04-02

Domain of

\sqrt{x}

function when solving

x^{2} - 2 = 0

Answer & Explanation

pypberissootcu

Beginner2022-04-03Added 14 answers

Step 1
There are two distinct concepts at play here. $\sqrt{x}$ is a function denoting the nonnegative square root of x.
However, the general solution to $a^{2} = b^{2}$ is $a = \pm b$
So $\sqrt{25} = 5$ , but if $x^{2} = 25$ then $x = \pm 5$ . As Gae. S. has already mentioned, the easiest way to see that $a^{2} = b^{2} \Leftrightarrow a = \pm b$ is through factorisation:
$a^{2} = b^{2}$
$\Leftrightarrow a^{2} - b^{2} = 0$
$\Leftrightarrow (a + b) (a - b) = 0$
$\Leftrightarrow a = - b$ or $a = b$
So, in your example, we have
$x^{2} = {(\sqrt{2})}^{2} \Leftrightarrow x = \pm \sqrt{2}$

armejantm925

Beginner2022-04-04Added 20 answers

Step 1
In general

\sqrt{x^{2}} = | x |

So your equation is equivalent to

| x | = \sqrt{2}

And from that last equation you find the two roots

x = - \sqrt{2}

and

x = \sqrt{2}

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