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

Step 1: Move constant term to the other side

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

Step 2: Factor out the coefficient of $x^2$

Factor out the 2 from the left-hand side: $2(x^2 + 6x) = 12$

Now divide both sides by 2: $x^2 + 6x = 6$

Step 3: Complete the square

To complete the square, take half of the coefficient of $x$, which is 6, divide by 2 to get 3, and then square it: $\left(\frac{6}{2}\right)^2 = 9$

Add 9 to both sides: $x^2 + 6x + 9 = 6 + 9$ $(x + 3)^2 = 15$

Step 4: Solve for $x$

Take the square root of both sides: $x + 3 = \pm \sqrt{15}$

Finally, subtract 3 from both sides: $x = -3 \pm \sqrt{15}$

Final Solution:

The solution to the equation is: $x = -3 \pm \sqrt{15}$

Would you like more details on any step?

Related Questions:

1. What are the steps for solving quadratic equations using the quadratic formula?
2. How can I use completing the square to derive the quadratic formula?
3. What is the geometric meaning of completing the square?
4. How can completing the square help in finding the vertex of a parabola?
5. How do you solve a quadratic equation when the discriminant is negative?

Tip:

Completing the square is a great method for converting a quadratic equation into vertex form, making it easier to graph or analyze geometrically.