Question 251829
The trick here lies in simplifying {{{sqrt(-32)}}}


{{{sqrt(-32)}}} Start with the given expression 



{{{sqrt(-1*32)}}} Factor out a negative 1



{{{sqrt(-1)*sqrt(32)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}



{{{i*sqrt(32)}}} Replace {{{sqrt(-1)}}} with {{{i}}} (remember {{{i=sqrt(-1)}}})



Now lets simplify {{{sqrt(32)}}}:

  


The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.



So let's list the factors of 32



Factors:

1, 2, 4, 8, 16, 32



Notice how 16 is the largest perfect square, so lets factor 32 into 16*2



{{{sqrt(16*2)}}} Factor 32 into 16*2
 
 
 
{{{sqrt(16)*sqrt(2)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
 
 
{{{4*sqrt(2)}}} Take the square root of the perfect square 16 to get 4 
 
 
 
This means that the expression {{{sqrt(32)}}} simplifies to {{{4*sqrt(2)}}}



So the expression {{{sqrt(-32)}}} simplifies to {{{4*i*sqrt(2)}}} (just reintroduce {{{i}}} back in)



In other words, {{{sqrt(-32)=4*i*sqrt(2)}}}



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Going back to {{{x=(-2+-sqrt(-32))/2}}}, this expression simplifies to {{{x=(-2+- 4i*sqrt(2))/2}}}. From there, it reduces to {{{x=-1+- 2i*sqrt(2)}}}



So the solutions are {{{x=-1+ 2i*sqrt(2)}}} or {{{x=-1- 2i*sqrt(2)}}}