Question

why do extraneous solutions happen

Answer 1

to what kind of equations?

Answer 2

to radicand equations

Answer 3

\[-2\sqrt{x + 3} = 8\]

Answer 4

like the one above

Answer 5

because if you square both sides, you can introduce a solution to the squared equation that is not a solution to the original equation
for example, the solution to \(x=4\) is \(\{4\}\) but the solution to \(x^2=16\) is \(x=4\) or \(x=-4\)

Answer 6

the one above \[-2\sqrt{x + 3} = 8\] has no solutions
the right hand sides \(8\) is a positive number, but the left hand side \(-2\sqrt{x+3}\) is always less than or equal to zero

Answer 7

I get it now
thank you for your help