Why does Solving system of quadratic equations gives extra roots

Question

Consider these system of Equations
\begin{align*}
\begin{cases}
x^2+4x+4=0\\\\
x^2+5x+6=0
\end{cases}
\end{align*}

For solving them
We have

Method 1-

Subtract both equations

So
$-x-2=0$

Hence,
$x=-2$

Method-2

Add both equations

$2x^2+9x+10=0$

After applying quadratic formula, we get

$x=-2$ or $x=-5/2$. But only $x=-2$ satisfies the system of equation.

Why is the $-5/2$ not satisfying the system of equations, what is intuition behind the error in method 2?

Answer 1

HINT

You can factor both polynomials according to your preferred method in order to obtain:

\begin{align*} \begin{cases} x^{2} + 4x + 4 = 0\\\\ x^{2} + 5x + 6 = 0 \end{cases} \Longleftrightarrow \begin{cases} (x+2)^{2} = 0\\\\ (x+2)(x+3) = 0 \end{cases} \end{align*}

Can you take it from here?