Completing the Square: A Visual Guide

Welcome to completing the square! Today we'll explore how to visualize quadratic expressions using algebra tiles.Let's start with the expression x squared plus six x.We'll use a grid to help us arrange our algebra tiles.The x squared term is represented by a blue square with sides of length x.The six x term is represented by six rectangular strips. Each strip has a length of x and a width of one unit.We'll arrange three strips horizontally along the top of our x squared tile.And three strips vertically along the side.Notice how we've split our six x strips into two groups of three. This arrangement will help us form a larger square in the next step.This arrangement leaves an empty space in the corner. In the next section, we'll see how to fill this space to complete our square.Now that we have our x squared tile and six x strips arranged, we need to complete the square shape.Notice the empty corner in our shape. To make this a perfect square, we need to fill this gap.Since we have three x strips on each side, we need three times three, or nine unit squares to fill the corner.Let's add these nine unit squares to complete our perfect square.Now our shape is a perfect square with sides of x plus three.Algebraically, we've added nine to x squared plus six x, creating a perfect square trinomial.This geometric model shows why we add the square of half the coefficient of x when completing the square.Now that we have our perfect square, let's convert it to standard form.The completed square shape x squared plus six x plus nine can be written as x plus three squared.But what if our original expression had a different constant term, like five instead of nine?Let's break this down step by step.First, we add and subtract the difference between nine and five, which is four.This allows us to group the perfect square terms while maintaining equality.The difference of four comes from subtracting our original constant, five, from the perfect square constant, nine.This difference of four must be subtracted to maintain the equality of our expression.