Completing the square is a key method for solving quadratic equations. However, students sometimes make mistakes that can be confusing and lead to wrong answers. Here are some common errors to watch out for, along with tips to fix them.

1. Forgetting the Coefficient of $x^2$

A common error happens when students forget about the number in front of $x^2$ (called the coefficient). The first step is to make sure that this number is 1.

If you have an equation like $ax^2 + bx + c$ (where $a$ is not equal to 1), make sure to factor out $a$ before trying to complete the square. If you skip this step, your calculations can go wrong.

Tip: Always divide each part of the equation by $a$ first:

$$y = ax^2 + bx + c \rightarrow y = a\left(x^2 + \frac{b}{a}x + \frac{c}{a}\right)$$

2. Incorrectly Finding Half of $b$

Another frequent mistake is getting half the number $b$ (the coefficient of $x$) wrong. Students sometimes forget to divide it by 2 or mix up fractions, which leads to an incorrect number that doesn't help in making the perfect square.

Tip: When you need to complete the square, make sure to carefully divide $b$ by 2:

$$\text{If } b = 6, \text{ then } \frac{b}{2} = \frac{6}{2} = 3.$$

3. Neglecting to Square the Half

After correctly finding half of $b$, the next step is to square this number. A common mistake is forgetting to square it or messing up the calculation. This can make the equation wrong.

Tip: Once you have half of $b$, always remember to square it. For example:

$$\left(\frac{b}{2}\right)^2 = 3^2 = 9.$$

4. Failing to Adjust the Constant Term

Usually, after you add the square term to the equation, students forget to adjust the constant term the right way. If you don’t do this, the equation will not match the original one.

Tip: Keep the equation balanced! If you add something on one side, you have to subtract or add the same thing to the other side. For example, if you add $9$, you must also subtract it from the constant:

$$y = (x + \frac{b}{2})^2 - k \text{ (where $k$ is calculated to keep the original equation true).}$$

5. Forgetting to Rewrite the Equation

One final mistake is not rewriting the equation in the completed square form. Students often make the necessary changes but forget to present it correctly as $(x + p)^2 + q$, where $p$ and $q$ come from their calculations.

Tip: Always take a moment to make sure that your final equation matches the completed square format and double-check your numbers.

In conclusion, while completing the square might seem tricky and filled with possible mistakes, knowing these common errors can make your work easier. Practice this method with organized exercises, check each step carefully, and don’t be afraid to ask for help. This will boost your confidence in solving quadratic equations!