what values if x are solutions of the equation. square root 1 - X squared plus 2 equals 3

Answer :

We have to find the values of x that are solution for the equation

[tex]\sqrt{1-x^2}+2=3[/tex]

To solve this equation, the first thing we will do is pass, the 2 that is on the left side, to the right side with the opposite sign

[tex]\begin{gathered} \sqrt{1-x^2}=3-2 \\ \\ \sqrt{1-x^2}=1 \end{gathered}[/tex]

Now we will take power two, on both sides of the equation

[tex]\begin{gathered} (\sqrt{1-x^2})^2=1^2 \\ \\ 1-x^2=1 \end{gathered}[/tex]

Now we will pass the x square to the rigth with opposite sign, and the 1 on the right to the left with opposite sign

[tex]\begin{gathered} 1-1=x^2 \\ \\ 0=x^2 \end{gathered}[/tex]

Finally, we take square root on both sides to find

[tex]\begin{gathered} \pm\sqrt{0}=\sqrt{x^2} \\ \\ \pm0=x \end{gathered}[/tex]

Where we have to include both square roots, the positive and the negative one. In this specific case, both are the same, because the solution is 0. Therefore, we conclude:

[tex]x=0[/tex]

We conclude that the provided equation has a unique solution, and that this solution is 0.