2.2. Completing the Square Method

1. Rearrange the equation: move the constant to the other side.

In this first step, we take a quadratic equation and isolate the term containing the variable on one side of the equation. We do this by moving the constant term (the number without the variable) to the other side. This sets the stage for applying the method of completing the square.

Imagine you are rearranging books on a shelf. You want to focus on a specific set of books (the variable term), so you take the irrelevant ones (the constant) and place them aside temporarily. This way, you can clearly see what you are working with.

2. Make the coefficient of 𝑥² equal to 1 (if necessary).

The next step involves ensuring that the coefficient (the number in front) of the x² term is equal to 1. If it is not, we can achieve this by dividing the entire equation by the current coefficient. This is crucial for the subsequent steps of completing the square.

Think of this step like scaling a recipe. If a recipe calls for 2 cups of flour but you only want to make a half batch, you divide everything by 2 to make it simpler. Here, we are simplifying the equation to make the next steps easier.

3. Add and subtract the square of half the coefficient of 𝑥.

In this step, we take the coefficient of the x term, divide it by 2, and then square that result. We add this squared value to one side of the equation and also subtract it to keep the equation balanced. This will help us form a perfect square trinomial.

Imagine you're adjusting a recipe again, but this time you're trying to achieve a specific flavor balance. You carefully add a bit of spice (the square) to enhance the flavor without altering the overall taste of the dish (keeping the equation balanced).

4. Rewrite as a perfect square and solve.

Finally, we rewrite the quadratic expression as a perfect square, which is in the form of (x + a)² = b. After forming this equation, we can take the square root of both sides and solve for x by isolating it. This provides us with the solutions to the quadratic equation.

Think of this step as finalizing your project. Just as you would present your project in a neat binder (the perfect square), you also check to ensure all parts are included (solving for x). This way, your work is complete and ready for submission.

Example: Solve 𝑥² + 6𝑥 + 5 = 0 by completing the square. Solution: 𝑥² + 6𝑥 = −5 Add 9 to both sides: 𝑥² + 6𝑥 + 9 = 4 (𝑥 + 3)² = 4 𝑥 + 3 = ±2 𝑥 = −1 or 𝑥 = −5

Let's solve the equation step by step. First, we rearrange it to get x² + 6x = -5. Then we ensure the coefficient of x² is 1, which it already is. Next, we take half of the 6 (which is 3), square it (which gives us 9), and add it to both sides. This gives us x² + 6x + 9 = 4, or (x + 3)² = 4. Taking the square root of both sides, we have x + 3 = ±2, leading to our two solutions: x = -1 or x = -5.

Imagine you're setting up a room by placing furniture (the x terms), but first, you need to clear some space (move the constant to the other side). Once everything is in position, you ensure there's enough space to fit each piece perfectly (the perfect square), resulting in a well-decorated room (the solutions of the equation).

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