Find the roots/solution of the following equations using complex the square​​

To solve the quadratic equation using the method of completing the square, I'll show you the steps using the example equation:

### Equation: \( x^2 - 8x + 7 = 0 \)

1. **Move the constant term to the other side of the equation:**

   \[
   x^2 - 8x = -7
   \]

2. **Complete the square:**
   - To complete the square, take half of the coefficient of \( x \) (which is \(-8\)), square it, and add it to both sides of the equation.
   - Half of \(-8\) is \(-4\), and \((-4)^2 = 16\).

   Add 16 to both sides:

   \[
   x^2 - 8x + 16 = -7 + 16
   \]

   Simplifying:

   \[
   (x - 4)^2 = 9
   \]

3. **Solve for \( x \) by taking the square root of both sides:**

   \[
   x - 4 = \pm \sqrt{9}
   \]

   \[
   x - 4 = \pm 3
   \]

4. **Solve for \( x \):**

   - For \( x - 4 = 3 \):
     \[
     x = 3 + 4 = 7
     \]

   - For \( x - 4 = -3 \):
     \[
     x = -3 + 4 = 1
     \]

So, the solutions are \( x = 7 \) and \( x = 1 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How does completing the square help in deriving the quadratic formula?
2. What are the advantages of using the completing the square method?
3. Can the method of completing the square be used for all quadratic equations?
4. How does completing the square apply to equations with complex coefficients?
5. How can completing the square be used in real-world applications, like physics or finance?

**Tip:** Completing the square is also a key step in deriving the formula for the vertex of a parabola \( y = ax^2 + bx + c \), where the vertex \( x \) is found at \( x = -\frac{b}{2a} \).

Discover how to solve quadratic equations using the completing the square method. Step-by-step solution provided for the equation x^2 - 8x + 7 = 0. Explore the advantages of this method and its real-world applications. Suitable for high school students and advanced learners.

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