Understanding Completing the Square

Completing the square is an algebraic technique that simplifies the manipulation of quadratic expressions and solving of quadratic equations.
The process involves converting a quadratic equation in standard form (ax^2 + bx + c) into a perfect square format ((x + d)^2 + e).
The target of this process is to rewrite the quadratic equation in such a way as to clearly identify the vertex (h, k) in the graph of the equation.

Steps in Completing the Square

Step 1: If the coefficient of x^2 is not 1, divide every term by the coefficient of x^2 to make it 1.
Step 2: Rearrange the equation so that the x^2 term and the x term are on one side of the equal sign, and the constant is on the other side.
Step 3: To complete the square on the left side, take half of the coefficient of the x term, square it, and add it to both sides of the equation.
Step 4: The left side of the equation is now a perfect square and can be written in the form (x + d)^2.
Step 5: Solve for the value of x, if needed, by taking the square root on both sides, remembering to consider both the positive and negative roots.

It is especially useful for finding the vertex of a parabola, which is represented by the equation y = a(x - h)^2 + k where (h, k) is the vertex.
Completing the square is an essential part of deriving the Quadratic formula.
This method provides another way of graphing quadratic functions and provides the foundation for understanding the graph of a quadratic function.