How to Complete the Square in Quadratic Equations

Completing the square is a powerful technique used to solve quadratic equations. It allows us to rewrite the equation in a way that makes it easier to find the values of x. Let's break down the process step by step using the equation $$x^2 + 6x = 2$$ as our example.

### Step 1: Rearranging the Equation
First, we want to get the constant term on the right side of the equation. In our case, we already have:
$$x^2 + 6x = 2$$

### Step 2: Identifying the Coefficient of x
Next, take the coefficient of x, which is 6 in this case. The goal is to form a perfect square trinomial on the left side. To do this, we will divide the coefficient by 2:
$$6 \\ 2 = 3$$

### Step 3: Squaring the Result
Now, we square the result from the previous step:
$$3^2 = 9$$

### Step 4: Adding to Both Sides
We add this squared number (9) to both sides of the equation:
$$x^2 + 6x + 9 = 2 + 9$$
This simplifies to:
$$x^2 + 6x + 9 = 11$$

### Step 5: Writing as a Perfect Square
The left side of the equation can now be factored into a perfect square:
$$(x + 3)^2 = 11$$

### Step 6: Solving for x
Now we can solve for x by taking the square root of both sides:
$$x + 3 = \\pm \\sqrt{11}$$
This gives us:
$$x = -3 \\pm \\sqrt{11}$$

### Real-World Applications
Completing the square is not just a theoretical exercise; it has practical applications in various fields. For example, in physics, it can be used to derive the vertex form of a projectile’s motion equation, allowing one to determine the maximum height reached. In engineering, it can aid in optimizing design parameters.

### Conclusion
Completing the square is a valuable algebraic skill that helps in solving quadratic equations and understanding their properties. If you encounter quadratic equations in your studies, remember this method as it will make your problem-solving process much smoother. For more resources and practice problems, check out the 'equations' section on our site.