How to Understand the Quadratic Formula Step-by-Step

Getting the quadratic formula can seem really hard, and many students find it confusing. A quadratic equation usually looks like this:

$$ax^2 + bx + c = 0$$

To find the quadratic formula, we use a method called completing the square. This method can be tricky but don’t worry! We’ll break it down step-by-step.

Step 1: Make It Simple

First, if the number in front of $x^2$ (which we call $a$) is not 1, you need to divide every part of the equation by $a$:

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

This part might feel confusing because you’re working with different numbers and fractions.

Step 2: Get the Constant Away

Next, you want to move the constant (the number with no $x$) to the other side of the equation. This looks like:

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

Step 3: Complete the Square

Now comes the tricky part! You take half of the number in front of the $x$ (which is $\frac{b}{a}$), square it, and add it to both sides. This means you do:

$$\left(\frac{b}{2a}\right)^2$$

Now, your equation changes to:

$$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$

Step 4: Solve for $x$

Now, you need to take the square root of both sides. Be careful! You have to think about both the positive and negative answers:

$$x + \frac{b}{2a} = \pm \sqrt{\text{what you got from the right side}}$$

To find $x$, just isolate it by moving things around:

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

Conclusion

After you rearrange everything, you end up with the quadratic formula:

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

This might seem like a lot to take in, but understanding these steps is important. They lead you to a useful tool in algebra!