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Completing the Square: A Step-by-Step Guide

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Completing the Square: A Step-by-Step Guide

In the realm of algebra, quadratic equations play a pivotal role. These equations, characterized by their highest power being 2, can be solved using various methods, one of which is completing the square. This technique is particularly useful when the equation cannot be readily factored.

Understanding the Concept

The essence of completing the square lies in transforming a quadratic expression into a perfect square trinomial, which is a trinomial that can be factored as (ax + b)^2 or (ax - b)^2. To achieve this, we manipulate the equation by adding a specific constant term to both sides.

Step-by-Step Guide

Let's illustrate this process with an example. Consider the quadratic equation x^2 + 6x - 7 = 0.

  1. Move the constant term to the right side of the equation:

    2. x^2 + 6x = 7

  2. Calculate (b/2)^2, where b is the coefficient of the x term:

    3. In this case, b = 6. So, (b/2)^2 = (6/2)^2 = 9.

  3. Add (b/2)^2 to both sides of the equation:

    4. x^2 + 6x + 9 = 7 + 9

  4. Factor the left side as a perfect square trinomial:

    5. (x + 3)^2 = 16

  5. Apply the square root property:

    6. x + 3 = ±√16

  6. Solve for x:

    7. x = -3 ± 4

    Therefore, the solutions to the equation are x = 1 and x = -7.

Key Points

  • The coefficient of the x^2 term must be 1 before you can complete the square. If it's not, divide the entire equation by that coefficient.
  • Completing the square is a powerful technique for solving quadratic equations, especially when factoring is not straightforward.
  • The process involves manipulating the equation to create a perfect square trinomial on one side, allowing you to easily solve for the variable.

Practice Makes Perfect

To master completing the square, it's essential to practice with various quadratic equations. Start with simple examples and gradually work your way up to more complex ones. Remember, the key is to follow the steps systematically and pay attention to the algebraic manipulations involved.

By understanding the concept and practicing diligently, you can confidently solve quadratic equations using the completing the square method. This technique is a valuable tool in your algebraic arsenal, empowering you to tackle a wide range of mathematical problems.

High School Math