Solve $x^2-2x-42=0$x2−2x−42=0​  by completing the square.

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To solve the equation $ x^2 - 2x - 42 = 0 $ by completing the square, follow these steps:

1. **Rearrange the equation:**
   \[
   x^2 - 2x = 42
   \]

2. **Complete the square:**
   - Take the coefficient of $ x $ (which is $-2$), divide it by 2, and square it:
     \[
     \left(-\frac{2}{2}\right)^2 = 1
     \]
   - Add this value to both sides:
     \[
     x^2 - 2x + 1 = 42 + 1
     \]
     \[
     x^2 - 2x + 1 = 43
     \]

3. **Rewrite the left side as a square:**
   \[
   (x - 1)^2 = 43
   \]

4. **Take the square root of both sides:**
   \[
   x - 1 = \pm \sqrt{43}
   \]

5. **Solve for $ x $:**
   \[
   x = 1 + \sqrt{43} \quad \text{and} \quad x = 1 - \sqrt{43}
   \]

Thus, the solutions are:
\[
x = 1 + \sqrt{43} \quad \text{and} \quad x = 1 - \sqrt{43}
\]

If you need further details or have any questions, feel free to ask! Here are some related questions you might consider:

1. What is the approximate value of $ \sqrt{43} $?
2. How would you solve the equation using the quadratic formula instead?
3. What does the graph of the equation $ y = x^2 - 2x - 42 $ look like?
4. Can you explain why completing the square works for solving quadratic equations?
5. What are the implications of having two real solutions in this quadratic equation?

**Tip:** Completing the square can also help in understanding the vertex form of a quadratic function.

Solve x^2 - 2x - 42 = 0 by completing the square.

Discover how to solve the quadratic equation x^2 - 2x - 42 = 0 using the method of completing the square. This solution includes detailed steps and relevant mathematical concepts. Ideal for students in Grades 7-9.

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