Quadratic formula and completing the square

The quadratic formula is used to find the solutions (roots) of any quadratic equation in the standard form $ax^2 + bx + c = 0$. The formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Completing the square is a method used to solve quadratic equations or to rewrite a quadratic expression in vertex form $a(x-h)^2 + k$. The goal is to transform an expression of the form $x^2 + bx$ into a perfect square trinomial $(x+d)^2$ by adding a specific constant term. The general steps are:

1. Move the constant term to the other side of the equation (if solving an equation).
2. Divide the entire equation by $a$ if $a \neq 1$.
3. Take half of the coefficient of the $x$ term ($b/2$), square it ($(b/2)^2$), and add it to both sides of the equation.
4. Factor the perfect square trinomial and simplify the other side.
5. Take the square root of both sides and solve for $x$.

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