Introduction:

Completing the square is an essential algebraic technique that’s useful for solving quadratic equations. It involves forming a perfect square trinomial from a given quadratic expression, which can help you find the solutions more easily. In this article, we will explore three different methods to complete the square.

Method 1: Standard Completing the Square Technique

1. Write down the quadratic equation in the form Ax^2 + Bx + C = 0.

2. Divide every term by A so that you have a leading coefficient of x^2.

3. Take half of B, square it, and add this value to both sides of the equation.

4. Express the left side as a squared binomial.

5. Use the square root method to solve for x.

Example:

x^2 – 6x + 3 = 0

x^2 – 6x + (-6/2)^2 = -3 + (-6/2)^2

x^2 – 6x + 9 = 6

(x-3)^2 = 6

Method 2: Vertex Form Technique

1. Write down the quadratic equation in the form y = Ax^2 + Bx + C.

2. Divide every term by A as before, so you have a leading coefficient of x^2.

3. Re-write the equation in vertex form: y = A(x-h)^2 + k by identifying how much (h) you need to shift horizontally and (k) vertically.

4. Use your vertex form equation to determine values for h and k, and use these values to derive your vertex coordinates (h,k).

Example:

y = x^2 – 6x + 3

y = (x-3)^2 – 9

Method 3: Diagonal Sum Method

1. Write down the quadratic equation in the form Ax^2 + Bx + C = 0.

2. Divide every term by A, as before, so you have a leading coefficient of x^2.

3. Look for a pair of factors for the product of A and C whose sum equals B.

4. Decompose the middle term into two terms with those factors as coefficients.

5. Complete the square and solve for x

Example:

x^2 – 6x + 5 = 0

x^2 – 5x – x + 5 = 0

(x-5)(x-1) = 0

Conclusion:

These three methods provide you with various options to complete the square for solving quadratic equations. Whether you choose the standard technique, vertex form method, or diagonal sum approach, mastering these techniques will improve your problem-solving skills in algebra. Practice and choose the method best suited to your needs based on each specific problem, and soon completing the square will become second nature to you!