Solving Quadratic Equations by Completing the Square

Overview

• Completing the Square is a method to solve quadratic equations.
• Assumes prior knowledge or review of the five steps for completing the square.

Steps to Solve by Completing the Square

Step 1: Move the Constant

• Move the constant to the right side of the equation.
• Example: Subtract 2 from both sides to get (x^2 + 5x).

Step 2: Form a Perfect Square Trinomial

• Add a constant to both sides to form a perfect square trinomial.
• Use formula: ((b/2)^2) where b is the coefficient of the middle term.
• Example: If b = 5, ((5/2)^2 = 25/4).
• Equation becomes (x^2 + 5x + 25/4 = -2 + 25/4).

Step 3: Factor the Perfect Square Trinomial

• The trinomial factors into binomial squares.
• The binomial has terms of (b/2).
• Example: Factor (x + 5/2) twice to get ((x + 5/2)^2).

Step 4: Simplify the Right Side

• Convert constants to have common denominators.
• Example: (-2 = -8/4), so equation becomes ((x + 5/2)^2 = 17/4).

Step 5: Solve for x

• Take the square root of both sides, remember the (\pm) for both solutions.
• Example: (x + 5/2 = \pm \sqrt{17}/2).
• Solve by isolating x: (x = -5/2 \pm \sqrt{17}/2).

Step 6: Identify Solutions

• Recognize square root of 17 does not simplify.
• Solutions are irrational numbers.
• Two real solutions: (-5/2 + \sqrt{17}/2) and (-5/2 - \sqrt{17}/2).

Conclusion

• Completing the square provides two real solutions for the equation.
• Upcoming examples will be shown in the next video.