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Solution for complex roots in polar coordinates

If the quadratic equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} with real coefficients has two complex roots — the case where b 2 − 4 a c < 0, {\displaystyle b^{2}-4ac<0,} requiring a and c to have the same sign as each other — then the solutions for the roots can be expressed in polar form as

x 1, x 2 = r (cos θ ± i sin θ), {\displaystyle x_{1},\,x_{2}=r(\cos \theta \pm i\sin \theta),}

where r = c a {\displaystyle r={\sqrt {\tfrac {c}{a}}}} and θ = cos − 1 (− b 2 a c). {\displaystyle \theta =\cos ^{-1}\left({\tfrac {-b}{2{\sqrt {ac}}}}\right).}