Completing the Square
Basics of Completing the Square

Completing the square is a method used to solve quadratic equations, convert quadratic functions to vertex form, and understand the graph of a quadratic function.

It involves rearranging a quadratic equation to create a perfect square trinomial from the quadratic and linear terms.
Steps to Complete the Square

Begin by ensuring your quadratic equation is in the form ax^2 + bx + c = 0. If ‘a’ is not 1, divide every term by ‘a’ to make it 1.

Move the constant term (c) to the other side of the equation.

Take half of the coefficient (number) of the ‘x’ term (b), square it, and add it to both sides of the equation.

The left side of the equation can now be written as a binomial squared: (x + h)^2, where h is half of the original ‘b’ value.

Solve for ‘x’ by taking the square root of both sides, remembering to consider both the positive and negative square roots.
Key Points to Remember

Completing the square only works directly on quadratics in the form x^2 + bx + c = 0. If the coefficient of x^2 is not 1, it must be made 1 by dividing every term by that coefficient.

After rearranging the equation, the left side becomes a perfect square: (x + h)^2.

Once the equation is in the form (x + h)^2 = k, solve for ‘x’ by taking the square root of both sides. Remember to consider both the positive and negative square roots.

This method is particularly useful for determining the turning point (vertex) of a quadratic function, and for sketching its graph. The vertex form of a quadratic is y = a(x  h)^2 + k where (h, k) are the coordinates of the vertex.