Write a solution to the quadratic equation a*x^2+b*x+c=0 in a latex format
The quadratic formula can be used to find the solutions to the quadratic equation:

\begin{equation*}
ax^2+bx+c=0
\end{equation*}

The formula is:

\begin{equation*}
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation*}

where $a$, $b$, and $c$ are coefficients of the quadratic equation.

Therefore, the solutions to the quadratic equation are:

\begin{equation*}
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}
\end{equation*}

\begin{equation*}
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}
\end{equation*}

Note that the expression inside the square root, $b^2 - 4ac$, is called the discriminant. If the discriminant is negative, the quadratic equation has no real solutions, but it may have complex solutions. If the discriminant is zero, the quadratic equation has one real solution. If the discriminant is positive, the quadratic equation has two real solutions.
Thank you. Now explain what completing the square is in the context of quadratic equations, using LaTeX format.
Completing the square is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The general steps for completing the square are:

1. Move the constant term, `c`, to the right-hand side of the equation.

2. Divide both sides of the equation by a, so that the coefficient of the `$x^2$` term is 1.

3. Add and subtract the square of half the coefficient of the `$x$` term, `$(b/2a)^2$`, to the left-hand side of the equation:
```
$$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0$$
```
4. Factor the left-hand side of the equation as a square of a binomial:
```
$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$
```
5. Take the square root of both sides of the equation:

```
$$x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$$
```
6. Simplify and solve for x:
```
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
```
This is the quadratic formula, which can be used to solve any quadratic equation of the form `$ax^2 + bx + c = 0$`. Completing the square is a useful technique for solving quadratic equations, and it can also be used to derive the quadratic formula.