How to Solve Quadratic Equations by Completing the Square( simplemethd11 Series 1)G)

 

Completing the Square Masterclass: 8 Examples

The 4 Step Rule (when a = 1)

x² + bx + c = 0

1. Move constant (c) to the right.
2. Add (b/2)² to both sides.
3. Factor left as (x + b/2)².
4. Take square root and solve for x.

Practice Examples

1. x² + 6x - 7 = 0

x² + 6x = 7 → x² + 6x + 9 = 7 + 9 → (x + 3)² = 16

x + 3 = ±4 → x = 1 or x = -7

2. x² - 8x + 12 = 0

x² - 8x = -12 → x² - 8x + 16 = -12 + 16 → (x - 4)² = 4

x - 4 = ±2 → x = 6 or x = 2

3. x² - 5x + 6 = 0

x² - 5x = -6 → x² - 5x + 25/4 = -6 + 25/4 → (x - 5/2)² = 1/4

x - 5/2 = ±1/2 → x = 3 or x = 2

4. x² - 4x - 5 = 0

x² - 4x = 5 → x² - 4x + 4 = 5 + 4 → (x - 2)² = 9

x - 2 = ±3 → x = 5 or x = -1

5. x² + 10x + 21 = 0

x² + 10x = -21 → x² + 10x + 25 = -21 + 25 → (x + 5)² = 4

x + 5 = ±2 → x = -3 or x = -7

6. x² + 3x - 10 = 0

x² + 3x = 10 → x² + 3x + 9/4 = 10 + 9/4 → (x + 3/2)² = 49/4

x + 3/2 = ±7/2 → x = 2 or x = -5

Advanced Examples (Non-Monic: a ≠ 1)

7. 2x² + 5x - 3 = 0 (Divide by 2)

x² + 2.5x - 1.5 = 0 → x² + 2.5x = 1.5 → x² + 2.5x + 1.5625 = 1.5 + 1.5625

(x + 1.25)² = 3.0625 → x + 1.25 = ±1.75 → x = 0.5 or x = -3

8. 3x² - 5x - 2 = 0 (Divide by 3)

x² - (5/3)x = 2/3 → x² - (5/3)x + 25/36 = 2/3 + 25/36 → (x - 5/6)² = 49/36

x - 5/6 = ±7/6 → x = 2 or x = -1/3