To solve the quadratic equation $2x^2 + 12x + 3 = 15$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side

First, subtract 15 from both sides: $2x^2 + 12x + 3 - 15 = 0$ $2x^2 + 12x - 12 = 0$ $2x^2 + 12x = 12$

Step 2: Divide by the coefficient of $x^2$

Next, divide the entire equation by 2 to simplify: $x^2 + 6x = 6$

Step 3: Complete the square

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides. The coefficient of $x$ is 6, so: $\left( \frac{6}{2} \right)^2 = 9$ Add 9 to both sides: $x^2 + 6x + 9 = 6 + 9$ $(x + 3)^2 = 15$

Step 4: Solve for $x$

Now, take the square root of both sides: $x + 3 = \pm \sqrt{15}$ $x = -3 \pm \sqrt{15}$

Thus, the solutions are: $x = -3 + \sqrt{15} \quad \text{or} \quad x = -3 - \sqrt{15}$

Would you like further details on any of the steps?

Here are 5 related questions you could explore:

1. How do you complete the square for a quadratic with a coefficient other than 1 in front of $x^2$?
2. What are other methods to solve quadratic equations apart from completing the square?
3. How do you find the vertex of a parabola using the completed square form?
4. How is completing the square related to the quadratic formula?
5. Can you always solve a quadratic equation using the method of completing the square?

Tip: Completing the square is a useful technique for converting a quadratic equation into a form that makes it easy to find the vertex or solve it.