How to Complete The Square

How to Complete The Square: A Step-by-Step Guide

Completing the square is a method used in algebra to solve quadratic equations. Here’s a step-by-step guide on how to do it:

**1. Start with a Quadratic Equation:

  • Begin with a quadratic equation in the form: ��2+��+�=0ax2+bx+c=0.

**2. Move the Constant to the Other Side:

  • If �c is positive, subtract �c from both sides of the equation. If �c is negative, add ∣�∣∣c∣ (the absolute value of �c) to both sides.
  • You should now have an equation of the form: ��2+��=−�ax2+bx=−c.

**3. Divide by the Coefficient of �2x2:

  • Divide both sides of the equation by �a to make the coefficient of �2x2 equal to 1.
  • The equation is now in the form: �2+���=−��x2+abx=−ac​.

**4. Take Half of the Coefficient of �x and Square It:

  • The term �2�2ab​ is half of the coefficient of �x. Square this value to get (�2�)2(2ab​)2.

**5. Add and Subtract (�2�)2(2ab​)2 to Both Sides:

  • Add (�2�)2(2ab​)2 to both sides to complete the square on the left-hand side.
  • The equation becomes: �2+���+(�2�)2=−��+(�2�)2x2+abx+(2ab​)2=−ac​+(2ab​)2.

**6. Factor the Left-Hand Side:

  • The left-hand side can be factored into (�+�2�)2(x+2ab​)2.
  • The equation now looks like: (�+�2�)2=−��+(�2�)2(x+2ab​)2=−ac​+(2ab​)2.

**7. Simplify the Right-Hand Side:

  • Simplify the right-hand side by performing the operations. This will give you a constant on the right.

**8. Take the Square Root of Both Sides:

  • Take the square root of both sides. Remember to consider both the positive and negative square roots.
  • You’ll get two possible values for �x: �=−�2�±−��+(�2�)2x=−2ab​±−ac​+(2ab​)2​.

**9. Simplify Further if Needed:

  • If possible, simplify the expression further.

**10. Check Your Solutions:

  • Plug your solutions back into the original quadratic equation to verify that they satisfy the equation.

Completing the square is a powerful technique for solving quadratic equations, especially when factoring or using the quadratic formula is not straightforward.

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