Completing the square is an important method in algebra. It's especially helpful when we want to use the Quadratic Formula. Let’s break this down so it’s easy to understand.

What is Completing the Square?

Completing the square is a way to change a quadratic equation, like $ax^2 + bx + c = 0$, into a perfect square trinomial. This makes the equation easier to solve.

Let’s look at a simple example:

$x^2 + 6x + 5 = 0.$

Our aim is to complete the square. We will focus on the $x^2 + 6x$ part. Here’s how we do it:

1. First, take half of 6: $6/2 = 3$.
2. Next, square it: $3^2 = 9$.

Now, we rewrite our equation. We’ll add and subtract this square:

$x^2 + 6x + 9 - 9 + 5 = 0.$

This simplifies to:

$(x + 3)^2 - 4 = 0.$

Now we can find the roots by isolating the square:

$(x + 3)^2 = 4.$

Taking the square root of both sides gives us:

$x + 3 = ±2,$

which leads to:

$x = -1 \quad \text{and} \quad x = -5.$

Why is Completing the Square Used for the Quadratic Formula?

So, why do we use completing the square to find the Quadratic Formula?

A quadratic is usually written as $ax^2 + bx + c = 0$. The first thing we do is divide the whole equation by $a$. This makes the numbers easier to work with:

$x^2 + \frac{b}{a}x + \frac{c}{a} = 0.$

Next, we move the constant to the other side:

$x^2 + \frac{b}{a}x = -\frac{c}{a}.$

Now, we complete the square. We take half of the $x$ coefficient, square it, and add it to both sides.

1. Half of $\frac{b}{a}$ is $\frac{b}{2a}$.
2. Squaring that gives us $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$.

When we add this to both sides, we have:

$x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}.$

Now the left side is a perfect square:

$\left(x + \frac{b}{2a}\right)^2 = \text{something on the right}.$

To make the right side easier, we need a common denominator. Eventually, we get to:

$x + \frac{b}{2a} = ±\sqrt{\text{something}}.$

Finally, isolating $x$ gives us:

$x = -\frac{b}{2a} ± \frac{\sqrt{b^2 - 4ac}}{2a}.$

This leads us to the famous Quadratic Formula:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}.$

Conclusion

In short, completing the square is very important for finding the Quadratic Formula. It changes a trinomial into a form that simplifies solving for $x$. This technique helps us find the roots easily and shows the structure of quadratic equations. By learning this process, you’ll be ready to solve quadratic equations, whether they’re simple or complicated. So next time you see a quadratic equation, remember how helpful completing the square can be!