Question

Answer 1

a: \(x+3+\sqrt{x^2-6x+9}\)

\(=x+3+\sqrt{\left(x-3\right)^2}\)

=x+3+3-x(x<=3)

=6

b: \(\sqrt{x^2+4x+4}-\sqrt{x^2}\)

\(=\sqrt{\left(x+2\right)^2}-\left|x\right|\)

=|x+2|-|x|

=x+2-(-x)=x+2+x=2x+2

c: \(\dfrac{\sqrt{x^2-2x+1}}{x-1}=\dfrac{\sqrt{\left(x-1\right)^2}}{x-1}=\dfrac{\left|x-1\right|}{x-1}\)

\(=\dfrac{x-1}{x-1}\left(x>1\right)\)

=1

d: \(\left|x-2\right|+\dfrac{\sqrt{x^2-4x+4}}{x-2}\)

\(=2-x+\dfrac{\sqrt{\left(x-2\right)^2}}{x-2}\)

\(=2-x+\dfrac{2-x}{x-2}\)

=2-x-1

=1-x