Completing the Square Method

Completing the square is a powerful method for solving quadratic equations. By transforming the equation into a perfect square trinomial, you can easily find the roots and understand the behavior of the quadratic function. Let's break it down step by step.

1. Identify the quadratic equation in standard form (ax² + bx + c = 0)

   • Ensure the equation is in the form where a, b, and c are constants.
   • Recognize that 'a' cannot be zero, as it would not be a quadratic equation.
   • Identify the coefficients to use in subsequent steps of completing the square.

2. Move the constant term to the right side of the equation

   • Rearrange the equation to isolate the variable terms on the left.
   • This step prepares the equation for manipulation without altering its equality.
   • Remember to change the sign of the constant when moving it across the equals sign.

3. Factor out the coefficient of x² if it's not 1

   • If 'a' (the coefficient of x²) is not 1, factor it out to simplify the equation.
   • This makes it easier to complete the square since the leading coefficient will be 1.
   • Ensure that the equation remains balanced by adjusting the right side accordingly.

4. Add the square of half the coefficient of x to both sides

   • Calculate half of 'b' (the coefficient of x), then square it.
   • Add this value to both sides of the equation to maintain equality.
   • This step creates a perfect square trinomial on the left side.

5. Factor the perfect square trinomial on the left side

   • Rewrite the left side as a squared binomial (e.g., (x + d)²).
   • This simplifies the equation and prepares it for the next steps.
   • Ensure that the binomial reflects the correct sign and value from the previous step.

6. Take the square root of both sides

   • Apply the square root to both sides of the equation.
   • Remember to consider both the positive and negative roots.
   • This step is crucial for finding the potential solutions for x.

7. Solve for x by isolating the variable

   • Rearrange the equation to isolate x on one side.
   • This may involve adding or subtracting terms from both sides.
   • Ensure to express the final solutions clearly, indicating both possible values.

8. Check your solutions in the original equation

   • Substitute the found values of x back into the original equation.
   • Verify that both sides of the equation are equal after substitution.
   • This step confirms the accuracy of your solutions and the validity of the method used.