1. The problem is to find the formula for completing the square. 2. Completing the square is a method used to convert a quadratic expression of the form $ax^2 + bx + c$ into a perfect square trinomial plus a constant. 3. The general formula for completing the square for $ax^2 + bx + c$ (assuming $a=1$ for simplicity) is: $$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$ 4. Explanation: - Take half of the coefficient of $x$, which is $\frac{b}{2}$. - Square it to get $\left(\frac{b}{2}\right)^2$. - Add and subtract this square inside the expression to form a perfect square trinomial. 5. This transforms the quadratic into a form that is easier to analyze or solve, especially for finding roots or vertex form of a parabola. 6. If $a \neq 1$, first factor out $a$ from the $x^2$ and $x$ terms before completing the square: $$ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$ This is the complete formula for completing the square.