Completing the Square

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.

Applying Completing the Square

  • 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.