Solving $3x^2 - 4x -2 = 0$ by completing the square I can't understand the solution from the textbook (Stroud & Booth's "Engineering Mathematics" on a problem that involves solving a quadratic equation by completing the square.
The equation is this:
$$
\begin{align}
3x^2 - 4x -2 = 0 \\
3x^2 - 4x = 2
\end{align}
$$
Now, divide both sides by three:
$$x^2 - \frac{4}{3}x = \frac{2}{3}$$
Next, the authors add to both sides the square of the coefficient of $x$, completing the square on the LHS:
$$x^2 - \frac{4}{3}x + \left(\frac{2}{3}\right)^2 = \frac{2}{3} + \left(\frac{2}{3}\right)^2$$
Now, the next two steps (especially the second step) baffle me. I understand the right-hand side of the first quation (how they get the value of $\frac{10}{9}$), but the last step is a complete mystery to me:
$$
\begin{align}
x^2 - \frac{4}{3}x + \frac{4}{9} = \frac{10}{9} \\
\left(x - \frac{2}{3}\right)^2 = \frac{10}{9}
\end{align}
$$
Can anyone please explain how they went from the first step to the second step?

Maybe it would be easier with prettier numbers? If you were to solve the equation $$x^2+200x+10000=0,$$ you’d recognize that $$(x+100)^2 = x^2+200x+10000$$ which would simplify things greatly, quickly leading to the solution $x=-100$.
If, however, your equation does not match the $(x+a)^2 = x^2 + 2ax + a^2$ formula perfectly, you'll have to add something to make it match. For example, to solve $$x^2+200x+9999=0,$$ you’ll have to transform it like this:
$$x^2 + 200x+9999+1-1=0$$
$$x^2 + 200x+10000-1=0$$
$$(x-100)^2-1=0$$
Note that $x^2+200x$ can only be completed by 10000 and not by any other number.
