Solving Quadratic Equations Using Completing the Square

Overview

• The focus is on solving quadratic equations using the completing the square method.
• The goal is to rearrange equations in the form of ((x + p)^2 + q).

Key Steps

1. Rearrange the Equation

   • Start with the quadratic equation and rearrange it into the form ((x + p)^2 + q).
   • This step was covered in a previous video.

2. Solving for x

   • Example 1: ( (x - 5)^2 - 7 = 0 )
     • Add 7 to both sides: ( (x - 5)^2 = 7 )
     • Square root both sides: ( x - 5 = \pm \sqrt{7} )
     • Add 5 to both sides: ( x = \pm \sqrt{7} + 5 )
     • Solutions:
       • ( x = \sqrt{7} + 5 )
       • ( x = -\sqrt{7} + 5 )
   • Example 2: ( (x + 3)^2 - 1 = 0 )
     • Add 1 to both sides: ( (x + 3)^2 = 1 )
     • Square root both sides: ( x + 3 = \pm 1 )
     • Subtract 3 from both sides: ( x = \pm 1 - 3 )
     • Solutions:
       • ( x = 1 - 3 = -2 )
       • ( x = -1 - 3 = -4 )

Key Points

• When square rooting, remember to consider both the positive and negative roots.
• Typically, the process results in two solutions for (x).
• Solutions can be converted into decimal form using a calculator if specified.

Conclusion

• Completing the Square Method:

  • Rearrange the quadratic equation into the form ((x + p)^2 + q).
  • Further rearrange to isolate (x) and solve.
  • The technique typically provides two solutions for quadratic equations.

• The video provides a clear and concise method for solving quadratic equations using the completing the square method.