Completing the square is a method used to solve quadratic equations by manipulating the equation to create a perfect square trinomial. This technique is especially helpful when the quadratic equation cannot be easily factored or when the quadratic coefficient is not equal to 1. Let's walk through the steps of completing the square.

1. Start with a quadratic equation in standard form: ax^2 + bx + c = 0, where a, b, and c are constants.

2. Divide the entire equation by a to make the coefficient of x^2 equal to 1: x^2 + (b/a)x + c/a = 0.

3. Move the constant term to the other side of the equation: x^2 + (b/a)x = -c/a.

4. Take half of the coefficient of x, square it, and add it to both sides of the equation. This step creates a perfect square trinomial on the left side: x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2.

5. Simplify the equation on both sides, factor the perfect square trinomial, and solve for x.

Example:

Let's solve the quadratic equation 2x^2 + 8x + 3 = 0 by completing the square.

1. Divide by 2: x^2 + 4x + 3/2 = 0.

2. Move the constant term to the other side: x^2 + 4x = -3/2.

3. Take half of 4 (the coefficient of x), which is 2, square it to get 4, and add it to both sides: x^2 + 4x + 4 = -3/2 + 4.

4. Simplify: (x + 2)^2 = 5/2.

5. Take the square root of both sides and solve for x: x + 2 = ±√(5/2). Therefore, x = -2 + √(5/2) or x = -2 - √(5/2).

Completing the square allows us to find the exact solutions of quadratic equations even if they are not easily factorable or have a quadratic coefficient that is not 1. Keep practicing this method to strengthen your understanding. You've got this!