Given a polynomial of the form:
[tex]x^2+bx+c[/tex]
The method of completing the square give us another expression for that polynomial:
Where k is given by:
[tex]k=c-\frac{b^2}{4}[/tex]
So first we have to identify b and c:
[tex]\begin{gathered} x^2+14x=-69 \\ x^2+14x+69=0 \\ x^2+14x+69=x^2+bx+c \end{gathered}[/tex]
Then b=14 and c=69. This means that by completing the square we get:
[tex]\begin{gathered} x^2+14x+69=(x+\frac{b}{2})^2+c-\frac{b^2}{4} \\ x^2+14x+69=(x+\frac{14}{2})^2+69-\frac{14^2}{4} \\ (x+\frac{14}{2})^2+69-\frac{14^2}{4}=(x+7)^2+69-49 \\ (x+7)^2+69-49=(x+7)^2+20 \end{gathered}[/tex]
So we have:
[tex]\begin{gathered} x^2+14x+69=(x+7)^2+20 \\ \text{And} \\ x^2+14x+69=0 \end{gathered}[/tex]
Which means that:
[tex]\begin{gathered} (x+7)^2+20=0 \\ (x+7)^2=-20 \\ x+7=\pm\sqrt[]{-20}=\pm i\sqrt[]{20}=\pm2i\sqrt[]{5} \\ x=-7\pm2i\sqrt[]{5} \end{gathered}[/tex]
Which means that the answer is the second option.
