While finding delta algebraically of quadratic functions, can we proceed in this way?

Question

while doing it this way, the answer obtained is wrong. Where have I possibly made an error?

Answer 1

Be careful when you solve equations with modulus. You didn't get the least upper boundary for $|x+2|$: $$ 0 \leq |x^2| < 4.5 \\ \Updownarrow \\ 0 \leq |x| < \sqrt{4.5} \\ \Updownarrow \\ -\sqrt{4.5} < x < \sqrt{4.5} \\ \Updownarrow \\ 2-\sqrt{4.5} < x+2 < 2+\sqrt{4.5} \\ \color{red}{\Uparrow} \\ 0 \leq |x+2| < |2-\sqrt{4.5}| \text{ and } 0 \leq |x+2| < |2+\sqrt{4.5}| \\ \Updownarrow \\ 0 \leq |x+2| < \min(|2-\sqrt{4.5}|, |2+\sqrt{4.5}|) = \sqrt{4.5}-2 $$