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.