Solve the quadratic equation by completing the square. *x^*2+14*x+* 42= 0

First, choose the appropriate form and fill in the blanks with the correct numbers. form options: option 1: (x+?)^2=? or option 2: (x-?)^2 =? solution: x=? 
Then, solve the equation. Simplify your answer as much as possible.
If there is more than one solution, separate them with commas.



To solve the quadratic equation \( x^2 + 14x + 42 = 0 \) by completing the square, let's go through the steps:

1. Start with the given equation:
   \[
   x^2 + 14x + 42 = 0
   \]

2. Move the constant term to the other side:
   \[
   x^2 + 14x = -42
   \]

3. To complete the square, take half of the coefficient of \(x\) (which is 14), divide it by 2, and square it:
   \[
   \left(\frac{14}{2}\right)^2 = 7^2 = 49
   \]

4. Add 49 to both sides of the equation to complete the square:
   \[
   x^2 + 14x + 49 = -42 + 49
   \]
   \[
   x^2 + 14x + 49 = 7
   \]

5. Now, the left-hand side is a perfect square trinomial:
   \[
   (x + 7)^2 = 7
   \]

   So, the appropriate form is:
   \[
   (x + 7)^2 = 7
   \]

6. Solve for \(x\) by taking the square root of both sides:
   \[
   x + 7 = \pm \sqrt{7}
   \]

7. Finally, solve for \(x\):
   \[
   x = -7 \pm \sqrt{7}
   \]

Thus, the solutions are:
\[
x = -7 + \sqrt{7}, \quad x = -7 - \sqrt{7}
\]

These are the two solutions for the quadratic equation.

Would you like further details or have any questions? 

Here are 5 related questions:
1. Can completing the square be used for any quadratic equation?
2. What other methods can solve quadratic equations?
3. How do we know when to use completing the square versus factoring?
4. What is the geometric interpretation of completing the square?
5. How can complex solutions arise when completing the square?

**Tip:** Completing the square is closely related to the vertex form of a parabola. Understanding this link can simplify graphing and interpreting quadratic equations!

Solve the quadratic equation by completing the square. x^2 + 14x + 42 = 0

Learn how to solve the quadratic equation x^2 + 14x + 42 = 0 using the completing the square method. This solution walks you through each step, covering key algebraic concepts such as perfect square trinomials and the square root property. Ideal for students in Grades 9-11.

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