Quantificational Logic: "For every number a, the equation $ax^2 + 4x -2 = 0$ has at least one solution if and only if $a \ge -2$."

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This is a solution check for my quantified representation of the following statement.

"For every number a, the equation $ax^2 + 4x -2 = 0$ has at least one solution if and only if $a \ge -2$."

$\forall a[$iff $a \ge -2$ there exists at least one solution such that $ax^2 + 4x -2 = 0]$

$\forall a[(a \ge -2) \leftrightarrow \exists x(ax^2 + 4x -2 = 0)]$