Using quantifiers to rewrite some simple mathematical statements

Question

Have I used quantifiers correctly to rewrite these sentences?

The equation $x^3=7$ has at least one root.

$∃x∈\mathbb R\;\;x^3-7=0$

The equation $x^2-2x-5=0$ has no rational roots.

non$(∀x∈\mathbb Q \implies x^2-2x-5=0)$

EDIT: Or how about $∀x∈\mathbb Q \implies x^2-2x-5≠0$ ?

Every equation on the form $x^3=a$ has one root.

$∀a∈\mathbb R \:\:∃x∈\mathbb R \;\;x^3=a$

Answer 1

The equation $x^2-2x-5=0$ has no rational roots.

non$(∀x∈\mathbb Q \implies x^2-2x-5=0)$

EDIT: Or how about $∀x∈\mathbb Q \implies x^2-2x-5≠0$ ?

1. The fragment $$∀x∈\mathbb Q\tag1$$ is informal and means $$\text{for every $x$ that is in $\mathbb Q$},$$ which is not a sentence.

2. Therefore, the sentence $$∀x \,\Big(x{∈}\mathbb Q \implies P(x)\Big)$$ can be abbreviated as $$∀x{∈}\mathbb Q \:\:P(x)\\\text{$P(x)$ holds for every rational $x$}.$$

3. Fragment $(1)$ means neither $$∀x \:\,x∈\mathbb Q\\\text{for every $x,\:x$ is rational}\\\text{everything is rational}$$ nor $$\text{every $x$ is rational}.$$

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non$(∀x∈\mathbb Q \implies x^2-2x-5=0)$

This isn't a coherent sentence. The modification $$\text{not $(∀x\,x∈\mathbb Q \implies x^2-2x-5=0)$}$$ means: everything is rational yet $x^2-2x-5\ne0.$

EDIT: Or how about $∀x∈\mathbb Q \implies x^2-2x-5≠0$ ?

This too isn't a coherent sentence. Based on the previous section, the correct answer is $$∀x \,\Big(x∈\mathbb Q \implies x^2-2x-5≠0\Big)$$ or, equivalently, $$∀x \,\Big(x^2-2x-5=0 \implies x\not∈\mathbb Q \Big)$$ or, equivalently, $$\lnot\exists x \,\Big(x^2-2x-5=0 \;\land\; x∈\mathbb Q \Big).$$