Factor with quadratic formula.
In general, given \(a{x}^{2}+bx+c\), the factored form is:
\[a(x-\frac{-b+\sqrt{{b}^{2}-4ac}}{2a})(x-\frac{-b-\sqrt{{b}^{2}-4ac}}{2a})\]
In this case, \(a=1\), \(b=4\) and \(c=13\).
\[(x-\frac{-4+\sqrt{{4}^{2}-4\times 13}}{2})(x-\frac{-4-\sqrt{{4}^{2}-4\times 13}}{2})\]
Simplify.
\[(x-\frac{-4+6\imath }{2})(x-\frac{-4-6\imath }{2})\]
\[\sqrt{(x-\frac{-4+6\imath }{2})(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Factor out the common term \(2\).
\[\sqrt{(x-\frac{-2(2-3\imath )}{2})(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Move the negative sign to the left.
\[\sqrt{(x-(-\frac{2(2-3\imath )}{2}))(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Cancel \(2\).
\[\sqrt{(x-(-(2-3\imath )))(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Remove parentheses.
\[\sqrt{(x-(-2+3\imath ))(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x-\frac{-4-6\imath }{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Factor out the common term \(2\).
\[\sqrt{(x+2-3\imath )(x-\frac{-2(2+3\imath )}{2})}+\sqrt{{x}^{2}+4x+5}=2of\]
Move the negative sign to the left.
\[\sqrt{(x+2-3\imath )(x-(-\frac{2(2+3\imath )}{2}))}+\sqrt{{x}^{2}+4x+5}=2of\]
Cancel \(2\).
\[\sqrt{(x+2-3\imath )(x-(-(2+3\imath )))}+\sqrt{{x}^{2}+4x+5}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x-(-2-3\imath ))}+\sqrt{{x}^{2}+4x+5}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{{x}^{2}+4x+5}=2of\]
Factor with quadratic formula.
In general, given \(a{x}^{2}+bx+c\), the factored form is:
\[a(x-\frac{-b+\sqrt{{b}^{2}-4ac}}{2a})(x-\frac{-b-\sqrt{{b}^{2}-4ac}}{2a})\]
In this case, \(a=1\), \(b=4\) and \(c=5\).
\[(x-\frac{-4+\sqrt{{4}^{2}-4\times 5}}{2})(x-\frac{-4-\sqrt{{4}^{2}-4\times 5}}{2})\]
Simplify.
\[(x-\frac{-4+2\imath }{2})(x-\frac{-4-2\imath }{2})\]
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x-\frac{-4+2\imath }{2})(x-\frac{-4-2\imath }{2})}=2of\]
Factor out the common term \(2\).
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x-\frac{-2(2-\imath )}{2})(x-\frac{-4-2\imath }{2})}=2of\]
Move the negative sign to the left.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x-(-\frac{2(2-\imath )}{2}))(x-\frac{-4-2\imath }{2})}=2of\]
Cancel \(2\).
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x-(-(2-\imath )))(x-\frac{-4-2\imath }{2})}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x-(-2+\imath ))(x-\frac{-4-2\imath }{2})}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x-\frac{-4-2\imath }{2})}=2of\]
Factor out the common term \(2\).
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x-\frac{-2(2+\imath )}{2})}=2of\]
Move the negative sign to the left.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x-(-\frac{2(2+\imath )}{2}))}=2of\]
Cancel \(2\).
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x-(-(2+\imath )))}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x-(-2-\imath ))}=2of\]
Remove parentheses.
\[\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}=2of\]
Divide both sides by \(2\).
\[\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2}=of\]
Divide both sides by \(o\).
\[\frac{\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2}}{o}=f\]
Simplify \(\frac{\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2}}{o}\) to \(\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2o}\).
\[\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2o}=f\]
Switch sides.
\[f=\frac{\sqrt{(x+2-3\imath )(x+2+3\imath )}+\sqrt{(x+2-\imath )(x+2+\imath )}}{2o}\]
f=(sqrt((x+2-3*IM)*(x+2+3*IM))+sqrt((x+2-IM)*(x+2+IM)))/(2*o)
