Answer:
[tex]6-3\sqrt{3}[/tex]
If you compare this to [tex]a+b\sqrt{3}[/tex], then [tex]a=6[/tex] and [tex]b=-3[/tex].
Step-by-step explanation:
We are given [tex]x=\sqrt{3}-1[/tex] and we are asked to find the value of [tex]x^2-x+1[/tex].
So we will need to find [tex]x^2[/tex].
[tex]x=\sqrt{3}-1[/tex]
Square both sides:
[tex]x^2=(\sqrt{3}-1)^2[/tex]
Expand using identity: [tex](x-a)^2=x^2-2ax+a^2[/tex].
[tex]x^2=(\sqrt{3})^2-2(\sqrt{3})(1)+1^2[/tex]
[tex]x^2=3-2\sqrt{3}+1[/tex]
[tex]x^2=4-2\sqrt{3}[/tex]
Let's go back to the full [tex]x^2-x+1[/tex].
[tex]x^2-x+1[/tex]
[tex](4-2\sqrt{3})-(\sqrt{3}-1)+1[/tex]
[tex]4-2\sqrt{3}-\sqrt{3}+1+1[/tex]
[tex]6-3\sqrt{3}[/tex]
If you compare this to [tex]a+b\sqrt{3}[/tex], then [tex]a=6[/tex] and [tex]b=-3[/tex].
