We have to solve this equation by completing the square.
We have two terms from the quadratic equation: x² + 14x, so we can add the third term as:
[tex]\begin{gathered} x^2+14x=-69 \\ x^2+2(7)x+7^2=-69+7^2 \end{gathered}[/tex]We add the same term to both sides of the equation to preserve the equality.
We now can continue solving the equation as:
[tex]\begin{gathered} x^2+2(7)x+49=-69+49 \\ (x+7)^2=-20 \\ x+7=\pm\sqrt{-20} \\ x=-7\pm\sqrt{-20} \end{gathered}[/tex]This equation does not have solutions for x for the set of real values, as we have a square root of a negative number.
We can express the solution as complex numbers as:
[tex]\begin{gathered} x=-7\pm\sqrt{-20} \\ x=-7\pm\sqrt{20}i \\ x=-7\pm2\sqrt{5}i \end{gathered}[/tex]Answer: the solutions for the equation are x = -7 - 2(√5)i and x = -7 + 2(√5)i.
