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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 useful technique for solving quadratic equations. It allows us to rewrite a quadratic equation in the form (x + a)^2 = b, which makes it easier to solve for x.

Steps for Completing the Square

Here are the steps involved in completing the square:

  1. Move the constant term to the right side of the equation. This will leave you with a quadratic expression on the left side of the equation.
  2. Take half of the coefficient of the x term, square it, and add it to both sides of the equation. This will complete the square on the left side of the equation.
  3. Factor the left side of the equation. The left side should now be a perfect square trinomial, which can be factored as (x + a)^2.
  4. Take the square root of both sides of the equation. Remember to include both positive and negative square roots.
  5. Solve for x.

Example

Let's solve the equation x^2 + 6x - 7 = 0 by completing the square.

  1. Move the constant term to the right side: x^2 + 6x = 7
  2. Take half of the coefficient of the x term, square it, and add it to both sides: x^2 + 6x + 9 = 7 + 9
  3. Factor the left side: (x + 3)^2 = 16
  4. Take the square root of both sides: x + 3 = ±4
  5. Solve for x: x = -3 ± 4

Therefore, the solutions to the equation x^2 + 6x - 7 = 0 are x = 1 and x = -7.

Key Points

  • Completing the square is a useful technique for solving quadratic equations.
  • The process involves adding a constant term to both sides of the equation to create a perfect square trinomial on the left side.
  • The square root property is used to solve for x after the left side has been factored.

Additional Resources

I hope this knowledge base has been helpful. If you have any questions, please feel free to ask!

quadratic equations