Solving Quadratic Equations by Completing the Square
Answer:
[tex]x = \frac{i\sqrt{11} -1}{2}\\[/tex] and [tex]x = -\frac{i\sqrt{11} +1}{2}\\[/tex]
Step-by-step explanation:
Given:
[tex]2x^2 +2x +6 = 0[/tex]
Solution:
[tex]2x^2 +2x +6 = 0 \\ 2x^2 +2x = -6 \\ x^2 + x = -3 \\ x^2 +x +\frac{1}{4} = -3 +\frac{1}{4} \\ (x +\frac{1}{2})^2 = -\frac{12}{4} +\frac{1}{4} \\ (x +\frac{1}{2})^2 = -\frac{11}{4} \\ x +\frac{1}{2} = ±\sqrt{-\frac{11}{4}} \\ x +\frac{1}{2} = ±\frac{i\sqrt{11}}{2}[/tex]
Solving for the positive square root:
[tex]x +\frac{1}{2} = \frac{i\sqrt{11}}{2} \\ x = \frac{i\sqrt{11}}{2} -\frac{1}{2} \\ x = \frac{i\sqrt{11} -1}{2}[/tex]
Solving for the negative square root:
[tex]x +\frac{1}{2} = -\frac{i\sqrt{11}}{2} \\ x = -\frac{i\sqrt{11}}{2} -\frac{1}{2} \\ x = \frac{-i\sqrt{11} -1}{2} \\ x = -\frac{i\sqrt{11} +1}{2}[/tex]
