Quadratic Functions and Equations

Completing the Square

Completing the square is a method used to solve quadratic equations of the form . This technique rewrites the quadratic expression so that one side of the equation becomes a perfect square trinomial, making it easier to solve for the variable using square root properties.

• Definition: Completing the square involves manipulating a quadratic equation so that it takes the form , where and are constants.

• Square Root Property: If , then .

• Perfect Square Trinomial: An expression of the form .

Methods for Solving Quadratic Equations

Quadratic equations can be solved using several methods. The main approaches include:

Steps for Completing the Square

To solve by completing the square:

1. Move the constant term to the other side: .

2. If , divide both sides by to make the coefficient of equal to 1.

3. Add to both sides to create a perfect square trinomial.

4. Rewrite the left side as a squared binomial: .

5. Take the square root of both sides and solve for .

Example: Solving by Completing the Square

Consider the equation .

• Step 1: Move constant to the other side:

• Step 2: Add to both sides:

• Step 3: Rewrite as a square:

• Step 4: Take square root:

• Step 5: Solve for : or

Practice Problems

• Problem 1: Solve by completing the square.

• Problem 2: Solve by completing the square.

• Problem 3: Solve by completing the square.

Summary Table: Completing the Square Steps

Additional info: Completing the square is also foundational for deriving the quadratic formula and for analyzing the vertex form of a parabola in graphing.