The method of completing the square is used to transform a quadratic expression into a perfect square trinomial. It is particularly helpful when solving quadratic equations or rewriting them in vertex form.

Here’s a step-by-step explanation of the completing the square process for a quadratic expression of the form:

$ax^2 + bx + c$

Formula Derivation (for standard $ax^2 + bx + c$):

1. Start with the quadratic equation: $ax^2 + bx + c = 0$

2. Divide through by $a$ (if $a \neq 1$): $x^2 + \frac{b}{a}x = -\frac{c}{a}$

3. Move the constant term to the other side: $x^2 + \frac{b}{a}x = -\frac{c}{a}$

4. Complete the square by adding $\left( \frac{b}{2a} \right)^2$ to both sides: $x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2$

5. Factor the left-hand side as a perfect square: $\left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2$

6. Solve for $x$ by taking the square root of both sides (if solving a quadratic equation): $x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + \left( \frac{b}{2a} \right)^2}$

7. Isolate $x$ to find the solution(s): $x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}$

This leads to the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example of Completing the Square:

For the quadratic equation $x^2 + 6x + 5 = 0$:

1. Start with $x^2 + 6x = -5$.
2. Add $\left(\frac{6}{2}\right)^2 = 9$ to both sides: $x^2 + 6x + 9 = -5 + 9$
3. Factor the left side: $(x + 3)^2 = 4$
4. Solve for $x$: $x + 3 = \pm 2 \quad \Rightarrow \quad x = -3 \pm 2$ Thus, $x = -1$ or $x = -5$.

Let me know if you want any more details, or if you have any questions.

5 Related Questions:

1. How do you convert a quadratic function to vertex form using completing the square?
2. What are the real-life applications of completing the square?
3. Can you use completing the square for equations with complex numbers?
4. How does completing the square help in integrating quadratic functions?
5. What are the differences between completing the square and the quadratic formula?

Tip:

Always remember to adjust the constant term carefully when dividing through by the coefficient of $x^2$ to avoid errors.