how to complete the square of a quadratic equation

To complete the square of a quadratic equation, you're transforming a quadratic expression into a perfect square trinomial, which can be factored easily. Here's the step-by-step process for completing the square for a quadratic equation of the form:

\[
ax^2 + bx + c = 0
\]

### Steps:

1. **Move the constant to the other side**:
   If the equation has a constant, move it to the other side by subtracting or adding it:
   \[
   ax^2 + bx = -c
   \]
   If \(a \neq 1\), first divide the entire equation by \(a\) to make the coefficient of \(x^2\) equal to 1:
   \[
   x^2 + \frac{b}{a}x = -\frac{c}{a}
   \]

2. **Add a value to both sides to complete the square**:
   Take the coefficient of \(x\) (which is \(\frac{b}{a}\)), divide it by 2, and square it. This will be the number you add to both sides of the equation:
   \[
   \left( \frac{\frac{b}{a}}{2} \right)^2 = \left( \frac{b}{2a} \right)^2
   \]
   Add this value to both sides of the equation:
   \[
   x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2
   \]

3. **Rewrite the left side as a perfect square**:
   The left side is now a perfect square trinomial, so you can factor it as:
   \[
   \left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2
   \]

4. **Solve for \(x\)**:
   Take the square root of both sides:
   \[
   x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + \left( \frac{b}{2a} \right)^2}
   \]
   Finally, isolate \(x\):
   \[
   x = -\frac{b}{2a} \pm \sqrt{-\frac{c}{a} + \left( \frac{b}{2a} \right)^2}
   \]

### Example:

Complete the square for \(x^2 + 6x + 5 = 0\):

1. Move the constant to the other side:
   \[
   x^2 + 6x = -5
   \]

2. Take half of 6, square it, and add to both sides:
   \[
   \left( \frac{6}{2} \right)^2 = 9
   \]
   Add 9 to both sides:
   \[
   x^2 + 6x + 9 = 4
   \]

3. Factor the left side:
   \[
   (x + 3)^2 = 4
   \]

4. Solve for \(x\):
   \[
   x + 3 = \pm 2
   \]
   So, \(x = -3 \pm 2\), giving solutions:
   \[
   x = -1 \quad \text{or} \quad x = -5
   \]

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Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How is completing the square related to the quadratic formula?
2. Can all quadratic equations be solved by completing the square?
3. What are the advantages of completing the square over factoring?
4. How does completing the square help in graphing quadratic functions?
5. How does completing the square apply to real-world problems?

**Tip**: When solving quadratics, completing the square is especially useful when the quadratic equation cannot be easily factored.

Learn how to complete the square of a quadratic equation with this comprehensive guide. Understand the step-by-step process, key formulas, and theorems involved in solving quadratic equations by completing the square. Ideal for students in Grades 9-12.

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