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Completing the Square: A Step-by-Step Guide

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Completing the Square: A Step-by-Step Guide

Completing the square is a powerful algebraic technique used to solve quadratic equations. It's a method that allows us to rewrite a quadratic equation in the form (x + a)² = b, making it easier to solve for x. This technique is particularly useful when factoring the quadratic equation directly is challenging.

Understanding the Concept

The key idea behind completing the square is to manipulate a quadratic expression in the form ax² + bx + c into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored as (x + a)², where 'a' is a constant.

Steps to Complete the Square

Follow these steps to complete the square for a quadratic equation in the form ax² + bx + c = 0:

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

    2. ax² + bx = -c

  2. Divide both sides by the coefficient of the x² term (a):

    3. x² + (b/a)x = -c/a

  3. Take half of the coefficient of the x term (b/a), square it, and add it to both sides of the equation:

    4. x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²

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

    5. (x + b/2a)² = -c/a + (b/2a)²

  5. Simplify the right side of the equation:

    6. (x + b/2a)² = (b² - 4ac)/4a²

  6. Take the square root of both sides of the equation:

    7. x + b/2a = ±√(b² - 4ac)/2a

  7. Solve for x:

    8. x = -b/2a ±√(b² - 4ac)/2a

Example

Let's solve the quadratic equation x² + 6x + 5 = 0 by completing the square.

  1. Move the constant term:

    2. x² + 6x = -5

  2. Take half of the coefficient of x, square it, and add it to both sides:

    3. x² + 6x + (6/2)² = -5 + (6/2)²

    x² + 6x + 9 = -5 + 9

  3. Factor the left side:

    4. (x + 3)² = 4

  4. Take the square root:

    5. x + 3 = ±2

  5. Solve for x:

    6. x = -3 ± 2

    Therefore, the solutions are x = -1 and x = -5.

Practice Questions

Try completing the square to solve these quadratic equations:

  1. x² - 4x - 5 = 0
  2. 2x² + 8x - 10 = 0
  3. 3x² - 12x + 9 = 0

Conclusion

Completing the square is a versatile technique that can be used to solve quadratic equations, find the vertex of a parabola, and even derive the quadratic formula. By understanding the steps and practicing with examples, you can master this valuable algebraic tool.

High School Math