Solving x² - 6x + 3 = 0 by Completing the Square

Completing the square is a powerful technique for solving quadratic equations.Let's start with the quadratic equation x squared minus six x plus three equals zero.This method helps us transform a quadratic equation into a perfect square trinomial, making it easier to solve.The first step is to move the constant term to the right side of the equation.Geometrically, we can think of x squared as the area of a square with side length x.The term negative six x represents the area of rectangles we need to subtract.Let's understand what each term represents in our equation.This arrangement allows us to focus on the terms containing x on the left side, preparing us for the next step of completing the square.Starting with our equation from the previous step:To create a perfect square, we first take half of the coefficient of x, which is negative six divided by two, giving us negative three.Next, we square this value. Negative three squared equals positive nine.We add and subtract this value, nine, on the left side of the equation. Adding nine helps create our perfect square, while subtracting nine maintains the equality.The first three terms, x squared minus six x plus nine, form a perfect square trinomial.This perfect square can be visualized as a geometric square with these terms:These terms can be factored as x minus three squared.Our equation is now x minus three squared, minus nine, equals negative three.This form will make it much easier to solve for x in our next step.Now that we have our perfect square trinomial, let's solve for x.First, let's add 9 to both sides to combine like terms.Next, we take the square root of both sides. Remember, when we take the square root, we need both positive and negative solutions.The square root and square cancel on the left side, giving us x minus 3 equals plus or minus the square root of 6.Finally, we add 3 to both sides to solve for x.Let's visualize these solutions on a number line.Our two solutions are x equals 3 plus the square root of 6, and x equals 3 minus the square root of 6.