Given [latex]ax^{2}+bx+c[/latex] with [latex]a\neq 0[/latex], follow these steps to complete the square:

1. Factor out [latex]a[/latex] from the quadratic and linear terms.
   [latex]ax^{2}+bx+c =a\big(x^{2}+\tfrac{b}{a}x\big)+c[/latex]

2. Add and subtract the square needed to complete the square inside the parentheses.
   The needed square is [latex]\left(\tfrac{b}{2a}\right)^{2}[/latex].
   [latex]a\Big(x^{2}+\tfrac{b}{a}x+\Big(\tfrac{b}{2a}\Big)^{2}-\Big(\tfrac{b}{2a}\Big)^{2}\Big)+c[/latex]

3. Regroup so the first three terms form a perfect square; keep the subtraction inside.
   [latex]a\Big(\big(x+\tfrac{b}{2a}\big)^{2}-\Big(\tfrac{b}{2a}\Big)^{2}\Big)+c[/latex]

4. Distribute [latex]a[/latex] and combine constants to finish in vertex form.
   [latex]\begin{align} a\big(x+\tfrac{b}{2a}\big)^{2}-a\Big(\tfrac{b}{2a}\Big)^{2}+c\\ &=a\big(x+\tfrac{b}{2a}\big)^{2}-\tfrac{b^{2}}{4a}+c\\ &=a(x-h)^{2}+k \end{align}[/latex]

Here
[latex]h=-\tfrac{b}{2a}[/latex] and [latex]k=c-\tfrac{b^{2}}{4a}[/latex].