Completing the square is a fundamental algebraic technique used to rewrite quadratic expressions. It transforms expressions like x squared plus six x plus five into the vertex form, such as x plus three squared minus four. This method reveals important features like the vertex of a parabola.
Let's break down the process into clear steps. First, start with your quadratic expression. Next, take half of the coefficient of x, which is six divided by two equals three. Then square this value to get nine. Add and subtract this value to maintain equality. Finally, factor the perfect square trinomial to get x plus three squared minus four.
When the leading coefficient is not one, we must factor it out first. For two x squared minus eight x plus ten, we factor out two from the first two terms. Then we complete the square inside the parentheses. Half of negative four is negative two, and negative two squared is four. We add four inside and subtract two times four equals eight outside. This gives us two times x minus two squared plus two, with vertex at two comma two.