Question

Why squaring both sides of a radical equation is an irreversible operation?

Answer 1

See explanation...

Explanation:

Given an equation to solve of the form:

`"left hand expression " = " right hand expression"`

we may attempt to simplify the problem by applying the same function `f(x)` to both sides to get:

`f(" left hand expression ") = f(" right hand expression ")`

Any solution of the original equation will be a solution of this new equation.

However, note that any solution of the new equation may or may not be a solution of the original one.

If `f(x)` is one to one - e.g. multiplication by a non-zero constant, cubing, adding or subtracting the same thing from both sides - then solutions of the new equation will be solutions of the original.

In the case of `f(x) = x^2`, we have a function which is not one to one. For example `f(-x) = f(x)`. So solutions of the new equation may not be solutions of the original one.

For example, given:

`sqrt(2x+1) = -sqrt(x+3)`

We can square both sides of the equation to get:

`2x+1 = x+3`

This new equation has solution `x=2`, but it is not a solution of the original equation.