Answer:
[tex]x_1 = -2 + 2i\\\\x_2 = -2 -2i[/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]7x ^ 2 + 28x +56 = 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 = 7[/tex]. Then:
[tex]7(x ^ 2 + 4x) = -56[/tex]
2. Take the coefficient b that accompanies the variable x. In this case the coefficient is 4. Then, divide by 2 and the result squared it.
We have:
[tex]\frac{4}{2} = 2\\\\(\frac{4}{2})^2 = 4[/tex]
3. Add the term obtained in the previous step on both sides of equality, remember to multiply by the common factor [tex]a = 7[/tex]:
[tex]7(x ^ 2 + 4x + 4) = -56 + 7(4)[/tex]
4. Factor the resulting expression, and you will get:
[tex]7(x + 2) ^ 2 = -28[/tex]
Now solve the equation:
Note that the term [tex]7(x + 2) ^ 2[/tex] is always [tex]> 0[/tex] therefore it can not be equal to -28.
The equation has no solution in real numbers.
In the same way we can find the complex roots:
[tex]7(x+2)^2 = -28\\\\(x+2)^2 = -\frac{28}{7}\\\\x+2 = \sqrt{-\frac{28}{7}}\\\\x = -2 + \sqrt{-4}\\\\x = -2 + 2\sqrt{-1}\\\\x_1 = -2 + 2i\\\\x_2 = -2 -2i[/tex]
