Answer:
[tex]x_1 = -1 + \sqrt{2}i\\\\x_1 = -1 - \sqrt{2}i[/tex]
Step-by-step explanation:
In this problem we have the equation of the following quadratic equation and we want to solve it using the method of square completion:
[tex]-3x ^ 2 -6x -9 = 0[/tex]
The steps are shown below:
For any equation of the form: [tex]ax ^ 2 + bx + c = 0[/tex]
1. If the coefficient a is different from 1, then take a as a common factor.
In this case [tex]a = -3[/tex]. Then:
[tex]-3(x ^ 2 + 2x) = 9[/tex]
2. Take the coefficient b that accompanies the variable x. In this case the coefficient is 2. Then, divide by 2 and the result squared it.
We have:
[tex]\frac{2}{2} = 1\\\\(\frac{2}{2})^2 = 1[/tex]
3. Add the term obtained in the previous step on both sides of equality, remember to multiply by the common factor [tex]a = -3[/tex]:
[tex]-3(x ^ 2 + 2x + 1) = 9 -3(1)[/tex]
4. Factor the resulting expression, and you will get:
[tex]-3(x + 1) ^ 2 = 6[/tex]
[tex](x + 1) ^ 2 = -\frac{6}{3}[/tex]
[tex](x + 1) ^ 2 >0[/tex]
Then, The equation has no solution in real numbers.
In the same way we can find the complex roots:
Now solve the equation:
[tex](x + 1) ^ 2 = -\frac{6}{3}\\\\x+1 = \sqrt{-2}\\\\x = -1 + \sqrt{-2}\\\\x_1 = -1 + \sqrt{2}i\\\\x_1 = -1 - \sqrt{2}i[/tex]
