Simplify \(1\times {x}^{2}\) to \({x}^{2}\).
\[ln={x}^{2}+x+{x}^{2}-x+1\]
Simplify \({x}^{2}+x+{x}^{2}-x+1\) to \(2{x}^{2}+1\).
\[ln=2{x}^{2}+1\]
Subtract \(1\) from both sides.
\[ln-1=2{x}^{2}\]
Divide both sides by \(2\).
\[\frac{ln-1}{2}={x}^{2}\]
Take the square root of both sides.
\[\pm \sqrt{\frac{ln-1}{2}}=x\]
Simplify \(\sqrt{\frac{ln-1}{2}}\) to \(\frac{\sqrt{ln-1}}{\sqrt{2}}\).
\[\pm \frac{\sqrt{ln-1}}{\sqrt{2}}=x\]
Rewrite \(ln-1\) in the form \({a}^{2}-{b}^{2}\), where \(a=0\) and \(b=1\).
\[\pm \frac{\sqrt{{0}^{2}-{1}^{2}}}{\sqrt{2}}=x\]
Simplify \({0}^{2}\) to \(0\).
\[\pm \frac{\sqrt{0-{1}^{2}}}{\sqrt{2}}=x\]
Simplify \({1}^{2}\) to \(1\).
\[\pm \frac{\sqrt{0-1}}{\sqrt{2}}=x\]
Simplify \(0-1\) to \(-1\).
\[\pm \frac{\sqrt{-1}}{\sqrt{2}}=x\]
Simplify \(\sqrt{-1}\) to \(\sqrt{1}\imath \).
\[\pm \frac{\sqrt{1}\imath }{\sqrt{2}}=x\]
Simplify \(\sqrt{1}\) to \(1\).
\[\pm \frac{1\times \imath }{\sqrt{2}}=x\]
Simplify \(1\times \imath \) to \(\imath \).
\[\pm \frac{\imath }{\sqrt{2}}=x\]
Switch sides.
\[x=\pm \frac{\imath }{\sqrt{2}}\]
x=IM/sqrt(2),-IM/sqrt(2)
