1. The problem is to complete the square for a quadratic expression such as $ax^2 + bx + c$. 2. Start with the quadratic expression in the form $x^2 + bx + c$ (assume $a=1$ for simplicity). 3. To complete the square, take half of the coefficient of $x$, which is $\frac{b}{2}$, and square it, resulting in $\left(\frac{b}{2}\right)^2$. 4. Add and subtract this square inside the expression: $$x^2 + bx + \left(\frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$ 5. Rewrite the expression as a perfect square trinomial and a constant term: $$\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$ 6. Simplify the constant terms to complete the square: $$\left(x + \frac{b}{2}\right)^2 + \left(c - \left(\frac{b}{2}\right)^2\right)$$ 7. This expression is the completed square form of the quadratic. Example: Complete the square for $x^2 + 6x + 5$. 1. Take half of the coefficient of $x$: $\frac{6}{2} = 3$. 2. Square it: $3^2 = 9$. 3. Add and subtract 9: $$x^2 + 6x + 9 - 9 + 5$$ 4. Rewrite as perfect square minus constants: $$\left(x + 3\right)^2 - 4$$ 5. Final answer: $$\left(x + 3\right)^2 - 4$$