Overview
This lesson demonstrates how to solve quadratic equations using the method of completing the square, illustrated with several examples.
Completing the Square Method
- Start by moving all constant terms to the right side of the equation.
- If the coefficient in front of x² is not 1, factor it out from the x² and x terms.
- Take half of the x coefficient, square it, and add it to both sides of the equation.
- The left side becomes a perfect square trinomial, which can be factored as (x ± number)².
- After factoring, solve for x by taking the square root of both sides.
- Remember to consider both the positive and negative square root values, yielding two possible solutions.
Example 1: x² + 4x = 5
- Half of 4 is 2; add (2)² = 4 to both sides: x² + 4x + 4 = 9.
- Factor left: (x + 2)² = 9.
- Square root both sides: x + 2 = ±3.
- Solutions: x = 1 and x = -5.
Example 2: 2x² - 12x - 7 = 0
- Move -7: 2x² - 12x = 7.
- Factor 2: 2(x² - 6x) = 7.
- Half of -6 is -3; square to get 9; add 2×9 = 18 to both sides: 2(x² - 6x + 9) = 25.
- Factor left: 2(x - 3)² = 25.
- Divide by 2: (x - 3)² = 25/2.
- Square root: x - 3 = ±5/√2 ⇒ x - 3 = ±(5√2)/2.
- Solutions: x = 3 ± (5√2)/2.
Example 3: 3x² - 5x = 10
- Factor 3: 3(x² - (5/3)x) = 10.
- Half of -5/3 is -5/6; square to get 25/36; add 3×(25/36) = 25/12 to both sides.
- 10 + 25/12 = 145/12.
- Remove 3 by multiplying both sides by 1/3: (x - 5/6)² = 145/36.
- Square root: x - 5/6 = ±√145/6.
- Solution: x = (5 ± √145)/6.
Key Terms & Definitions
- Quadratic equation — An equation of the form ax² + bx + c = 0.
- Completing the square — A method to rewrite a quadratic in the form (x + p)² = q.
- Perfect square trinomial — A trinomial that factors into (x ± a)².
Action Items / Next Steps
- Practice solving quadratic equations by completing the square.
- Review factoring and working with square roots.