**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+��+�=0
*a**x*2+*b**x*+*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+��=−�
*a**x*2+*b**x*=−*c*.

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

- Divide both sides of the equation by �
*a*to make the coefficient of �2*x*2 equal to 1. - The equation is now in the form: �2+���=−��
*x*2+*a**b**x*=−*a**c*.

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

- The term �2�2
*a**b* is half of the coefficient of �*x*. Square this value to get (�2�)2(2*a**b*)2.

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

- Add (�2�)2(2
*a**b*)2 to both sides to complete the square on the left-hand side. - The equation becomes: �2+���+(�2�)2=−��+(�2�)2
*x*2+*a**b**x*+(2*a**b*)2=−*a**c*+(2*a**b*)2.

**6. **Factor the Left-Hand Side**:

- The left-hand side can be factored into (�+�2�)2(
*x*+2*a**b*)2. - The equation now looks like: (�+�2�)2=−��+(�2�)2(
*x*+2*a**b*)2=−*a**c*+(2*a**b*)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�)2*x*=−2*a**b*±−*a**c*+(2*a**b*)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.