Solving Quadratic Equations with Complex Solutions

Quadratic Equation Example

• Equation: 6x² - 2x = -3
• Objective: Solve using the quadratic formula

Steps to Solve

1. Set Equation to Zero

• Add 3 to both sides:
  • New Equation: 6x² - 2x + 3 = 0

2. Identify Coefficients

• A (leading coefficient) = 6
• B (second coefficient) = -2
• C (constant term) = 3

3. Apply Quadratic Formula

• Formula: x = [-b ± √(b² - 4ac)] / 2a
• Substitution:
  • x = [-(-2) ± √((-2)² - 4 * 6 * 3)] / (2 * 6)

4. Simplify the Solutions

• Calculate:
  • x = [2 ± √(4 - 72)] / 12
  • x = [2 ± √(-68)] / 12
• Recognize complex numbers (√ of negative value)

Simplification of Complex Solutions

5. Work with the Imaginary Part

• Split -68 as: -1 * 68
  • Recognize: √(-1) = i (imaginary unit)
  • x = [2 ± i√68] / 12

6. Further Simplify Square Root

• Split 68 into product of a perfect square:
  • 68 = 4 * 17
  • √4 = 2
• Substitute back:
  • x = [2 ± 2i√17] / 12

7. Final Simplification

• Factor out common term (2) in the numerator:
  • x = [2(1 ± i√17)] / 12
• Simplify division:
  • x = (1 ± i√17) / 6

Final Solutions

• Complex Number Solutions:
  • x₁ = (1 + i√17) / 6
  • x₂ = (1 - i√17) / 6

Conclusion

• Solved quadratic equation has two complex solutions.