Completing the Square Made Easy

Completing the square is a helpful way to solve quadratic equations. It works great when factoring the equation is tricky or when using the quadratic formula is complicated. Here’s how to do it step by step for a standard quadratic equation in the form (ax^2 + bx + c = 0), where (a), (b), and (c) are numbers.

Step-by-Step Guide

1. Make the Coefficient of (x^2) Equal to 1:

   • If (a) is not 1, divide the whole equation by (a). This changes the equation to (x^2 + \frac{b}{a} x + \frac{c}{a} = 0).

2. Rearrange the Equation:

   • Move the number without (x) (the constant) to the other side: [ x^2 + \frac{b}{a} x = -\frac{c}{a} ]

3. Find the Value to Complete the Square:

   • Take (\frac{b}{a}), divide it by 2, and then square it. This gives you the “square term”: [ \text{Square term} = \left(\frac{\frac{b}{a}}{2}\right)^2 = \frac{b^2}{4a^2} ]

4. Add and Subtract the Square Term:

   • Add this square term to both sides of the equation to keep it balanced: [ x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} ]

5. Rewrite the Left Side as a Square:

   • The left side can be written as a perfect square: [ \left(x + \frac{b}{2a}\right)^2 ]

6. Simplify the Right Side:

   • Combine the numbers on the right side: [ -\frac{c}{a} + \frac{b^2}{4a^2} = \frac{-4ac + b^2}{4a^2} ]

7. Set the Equation Equal:

   • Now your equation looks like this: [ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} ]

8. Take the Square Root of Both Sides:

   • Find (x) by taking the square root: [ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} ]
   • This means: [ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} ]

9. Final Solution:

   • Combine everything to get the solutions: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

Conclusion

Completing the square is a useful technique for solving quadratic equations. It helps to uncover important details about the quadratic function we are working with. Not only does it help us find solutions, but it can also show us the vertex (the highest or lowest point) of the parabola that represents the equation. Learning this method is crucial for building a strong math foundation, especially for algebra and calculus.

Key Facts

• Quadratic equations appear often in math problems. Around 25% of algebra problems for 8th graders are about them.
• The quadratic formula comes from completing the square, showing how important this method is for more advanced math.

Completing the square is an important skill to have. It adds a powerful tool to help solve math problems!