Question

Calculate:
(x+2(x-1)^(1/2))^(1/2)+(x-2(x-1)^(1/2))^(1/2)
for x>2 and for 1>=x>=2

Answer 1

\[\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\]

Answer 2

square the binomial:
\[
a^2=(x+\cancel{2\sqrt{x-1}})+2\sqrt{(x+2\sqrt{x-1})(x-2\sqrt{x-1})}+(x-\cancel{2\sqrt{x-1}})\\
a^2=2x+2\sqrt{x^2-4(x-1)}\\
a^2=2x+2\sqrt{x^2-4x+4}\\
a^2=2x+2\sqrt{(x-2)^2}\\
a^2=2x+2(x-2)\\
a^2=2x+2x-4\\
a^2=4(x-1)\\
\Large\boxed{a=\pm2\sqrt{x-1}}
\]
but we choose only the positive value since the given square-root problem is always positive.

Answer 3

formulae use:
\[
(a+b)^2=a^2+2ab+b^2\\
(a+b)(a-b)=a^2-b^2
\]

Answer 4

did you follow?

Answer 5

yep
the only problem is that this is actually a problem of absolute value of a real number

Answer 6

right,
|a|

Answer 7

when u take the square off the root u should absolute it, because this is like for 1>=x>=2 a=2 and for x>2

Answer 8

explain that
the only condition needed here would be
\(x\ge1\)
your other condition does not make sense.
\(1\ge x\ge2\)
means that x is less than or equal to 1 AND greater than or equal to 2 -> an illogical statement.

Answer 9

if x<1,
the radicals become undefined.

Answer 10

its the oposite 2>=x>=1

Answer 11

the answer to this problem is for 2>=x>=1 the expression is 2 and for x>2 the expression becomes 2(x-1)^(1/2)

Answer 12

right, that is looking at the domain
\[
a) \;x-1\ge0\implies x\ge 1\\
b)\;
x+2\sqrt{x-1}\ge0\qquad x-2\sqrt{x-1}\ge0\\
\implies x\ge 2\sqrt{x-1}\\
\implies x^2\ge 4(x-1)\\
\implies x^2-4x+4\ge0\\
\implies |x-2|\ge0\\
\implies x-2\ge 0\\
\implies x\ge 2
\]
resolving,
\[
a^2=(x+\cancel{2\sqrt{x-1}})+2\sqrt{(x+2\sqrt{x-1})(x-2\sqrt{x-1})}+(x-\cancel{2\sqrt{x-1}})\\
a^2=2x+2\sqrt{x^2-4(x-1)}\\
a^2=2x+2\sqrt{x^2-4x+4}\\
a^2=2x+2\sqrt{(x-2)^2}\\
a^2=2x+2\color{red}{|x-2|}\\
a^2=2x+2x-4\quad(\forall x\ge2)\qquad a^2=2x-2x+4\quad (\forall 1\le x\le2)\\
a^2=\cases{4(x-1)\qquad \forall \;x\ge2\\ 4\qquad \forall\;1\le x\le2}\\
\Large\boxed{a=\cases{2\sqrt{x-1}\quad\forall\; x>2\\2\qquad\qquad \forall\;1\le x\le2}}
\]

Answer 13

Thx a lot man, sorry for the confusion!!!

Answer 14

yw