Respuesta :
We need to add and subtract the same number in the expression so we can write it as a square and a constant, as in the following expression:
[tex]a\left(x-h\right)^2+k[/tex]The given expression is:
[tex]x^2-2x-2[/tex]In order to find the number that must be added and subtracted, let's expand the general expression:
[tex]\begin{gathered} a(x-h)^{2}+k \\ \\ a(x^2-2hx+h^2)+k \\ \\ ax^2-2ahx+ah^2+k \end{gathered}[/tex]Comparing the coefficients of the general and the given expression, we have:
[tex]\begin{gathered} x^{2}-2x-2 \\ \\ ax^{2}-2ahx+ah^{2}+k \\ \\ x^2=ax^2\Rightarrow a=1 \\ \\ -2x=-2ahx=2hx\Rightarrow h=1 \\ \\ -2=ah^2+k=1+k\Rightarrow k=-3 \end{gathered}[/tex]So, using a = 1, h = 1, and k = -3, we can write:
[tex]x^2-2x-2=1(x-1)^2-3=(x-1)^2-3[/tex]Notice that we obtain the same result by adding 3-3 to the original expression:
[tex]x^2-2x-2=x^2-2x-2+3-3=(x^2-2x+1)-3=(x-1)^2-3[/tex]Therefore, the answer is:
[tex]\begin{equation*} (x-1)^2-3 \end{equation*}[/tex]