If vx = - 2 does x= 4 or it have a solution because the root can never be negative

Question

My teacher says it has no solution 

Answer 1

If √x = - 2 does x= 4?

 

if we are dealing with Real numbers, then the square root of a negative number DOES NOT EXIST.

 

If we deal with the square root of a non-negative number, then its PRINCIPAL SQUARE ROOT is zero or some positive Real number, and furthermore it's IRRATIONAL unless we're taking the square root of a perfect square (such as 4 or 9, whose square roots are 2 and 3, respectively).

 

The square root of a negative number can be defined as i times the square root of its absolute value--this gives an Imaginary number as the square root of a negative number.

 

Now, I will discuss Michael J's answer, which is pasted below:

If you square both sides of the equation, you get

√(x)² = (-2)²

x = (-2) * (-2)

x = 4  this is incorrect, because the original statement is not satisfied by substituting four -- the square root of four is two, and only two.  ANY TIME YOU SQUARE AN EQUATION in order to derive an answer, you MUST check that answer; when you do so, you must rule out 4 because it is EXTRANEOUS. 

 

Answer 2

√(x)=-2  does have a solution.  If you square both sides of the equation, you get

 

√(x)² = (-2)²

 

   x = (-2) * (-2)

 

   x = 4

 

 

 

However, if you had an equation such that    √(-2) = x  , then x will have no real solution.  This is because you cannot take the square-root of a negative number.  In fact, you cannot take the nth root of a negative number, with n being positive.   This is because any number (either positive or negative) raised to an even power is ALWAYS positive.

 

 

For example:

 

2 * 2 = 4

 

(-2) * (-2) = 4