 completing the square addition. Consider the expression x squared plus 6x. We can think geometrically about this as a square of side x and a rectangle of side x and say width x and length 6. Now if we cut this rectangle in half and put one of those strips here, all the dimensions of this strip are 3 times x and we put the other strip here, 3x. We have almost formed a new square, a larger square of dimensions x plus 3. But we need a little piece. We need a little square with dimensions 3 times 3. That is, we need nine square units. So now we have made a square with dimensions x plus 3. How does this relate to this? Well, x squared plus 6x. When we add nine square units to it, this is equal to x plus 3 squared. So can we use this to solve a quadratic equation? Consider this one. This looks very similar to the previous expression, although now it's an equation and we have this one here. Let us first move that one over to the other side and we know that we can complete this expression. We can complete the square by adding 9 to it. That is 6 divided by 2 is 3. 3 squared is 9 and since we added that to the left hand side, we added to the right hand side of this equation and we have that x plus 3 squared is equal to 8. If we take the square root on both sides, we get that x plus 3 is equal to plus or minus the square root of 8. That is, x is equal to negative 3 plus or minus the square root of 8 is 2 square root of 2. So we have two values for x. x is negative 3 plus 2 square root of 2, or x can be negative 3 minus 2 times the square root of 2 and those are the solutions to this quadratic equation which we have solved by applying the technique called completing the square.