Let $n = [x]$, where $n$ is an integer and $n \leq x < n+1$.

Then the equation becomes:
$$4x^2 - 40n + 51 = 0$$
which gives
$$4x^2 = 40n - 51$$
$$x^2 = 10n - 12.75$$
$$x = \pm \sqrt{10n - 12.75}$$
But $n \leq x < n+1$, so $x$ must be real, i.e., $10n - 12.75 \geq 0$, or $n \geq 1.275$. Since $n$ is integer, $n \geq 2$.

Also, $x$ must satisfy $n \leq x < n+1$.

Let us consider $x = \sqrt{10n - 12.75}$ first.

We require:
$$n \leq \sqrt{10n - 12.75} < n+1$$

Let us solve $n \leq \sqrt{10n - 12.75}$:
$$n^2 \leq 10n - 12.75$$
$$n^2 - 10n + 12.75 \leq 0$$
The roots of $n^2 - 10n + 12.75 = 0$ are:
$$n = \frac{10 \pm \sqrt{100 - 4 \times 12.75}}{2} = \frac{10 \pm \sqrt{100 - 51}}{2} = \frac{10 \pm 7}{2}$$
So $n = \frac{17}{2} = 8.5$ or $n = \frac{3}{2} = 1.5$.

So $1.5 \leq n \leq 8.5$, i.e., $n = 2,3,4,5,6,7,8$.

Now, for each $n$ in $2$ to $8$, check if $\sqrt{10n - 12.75} < n+1$:

$\sqrt{10n - 12.75} < n+1$

Square both sides:
$$10n - 12.75 < (n+1)^2 = n^2 + 2n + 1$$
$$0 < n^2 - 8n + 13.75$$
The roots of $n^2 - 8n + 13.75 = 0$ are:
$$n = \frac{8 \pm \sqrt{64 - 55}}{2} = \frac{8 \pm 3}{2} = 5.5, 2.5$$
So $n < 2.5$ or $n > 5.5$

But $n \geq 2$, so $n = 2$ or $n > 5.5$, i.e., $n = 6,7,8$.

So possible $n$ are $2,6,7,8$.

Now, for each $n$ in $2,6,7,8$, $x = \sqrt{10n - 12.75}$, and $n \leq x < n+1$.

Check for $n = 2$:
$10 \times 2 - 12.75 = 20 - 12.75 = 7.25$
$\sqrt{7.25} \approx 2.6926$
$2 \leq 2.6926 < 3$ is true.

For $n = 6$:
$10 \times 6 - 12.75 = 60 - 12.75 = 47.25$
$\sqrt{47.25} \approx 6.8777$
$6 \leq 6.8777 < 7$ is true.

For $n = 7$:
$10 \times 7 - 12.75 = 70 - 12.75 = 57.25$
$\sqrt{57.25} \approx 7.5697$
$7 \leq 7.5697 < 8$ is true.

For $n = 8$:
$10 \times 8 - 12.75 = 80 - 12.75 = 67.25$
$\sqrt{67.25} \approx 8.2006$
$8 \leq 8.2006 < 9$ is true.

So for $n = 2,6,7,8$, $x = \sqrt{10n - 12.75}$ is a solution.

Now, consider $x = -\sqrt{10n - 12.75}$.

We require $n \leq x < n+1$, but $-\sqrt{10n - 12.75} < 0$, so $n < 0$.

But $n \geq 2$, so there are no solutions for $x = -\sqrt{10n - 12.75}$.

Therefore, the real solutions are:
$$x = \sqrt{10n - 12.75}, \quad n = 2,6,7,8$$
That is,
$$x_1 = \sqrt{7.25} \approx 2.6926$$
$$x_2 = \sqrt{47.25} \approx 6.8777$$
$$x_3 = \sqrt{57.25} \approx 7.5697$$
$$x_4 = \sqrt{67.25} \approx 8.2006$$

So all real solutions are $x = \sqrt{7.25}$, $x = \sqrt{47.25}$, $x = \sqrt{57.25}$, $x = \sqrt{67.25}$.