To complete the square for a quadratic equation like ( ax^2 + bx + c = 0 ), follow these steps:

Step 1: Move the Constant Term

First, we want to get the constant term ( c ) to one side of the equation. So, we move it to the right side:

[ ax^2 + bx = -c ]

Step 2: Make the Coefficient of ( x^2 ) Equal to 1

If ( a ) is not 1, divide everything by ( a ) to make the number in front of ( x^2 ) equal to 1:

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

Step 3: Find the Number to Complete the Square

Now, take half of the number in front of ( x ) (which is ( b/a )) and then square it. You can use this formula:

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

Step 4: Add and Subtract This Number

Add and subtract that squared number to the left side of the equation:

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

Step 5: Rewrite the Left Side

Now, the left side can be written as a perfect square:

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

Step 6: Move the Subtracted Square to the Right Side

Next, move the subtracted square to the right side:

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

Step 7: Solve for ( x )

Finally, take the square root of both sides to find ( x ):

[ x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + \left( \frac{b}{2a} \right)^2} ]

So, we find:

[ x = -\frac{b}{2a} \pm \sqrt{-\frac{c}{a} + \left( \frac{b}{2a} \right)^2} ]

By completing the square, you've rewritten the quadratic in a way that makes it easier to solve and understand what it means.