Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Find the answer

The equation x2+2x=ix^{2}+2 x=i has two complex solutions. Determine the product of their real parts.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The solutions are x+1=±eiπ824x+1= \pm e^{\frac{i \pi}{8}} \sqrt[4]{2}. The desired product is then (−1+cos⁡(π8)24)(−1−cos⁡(π8)24)=1−cos⁡2(π8)2=1−(1+cos⁡(π4))22=1−22\left(-1+\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)\left(-1-\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)=1-\cos ^{2}\left(\frac{\pi}{8}\right) \sqrt{2}=1-\frac{\left(1+\cos \left(\frac{\pi}{4}\right)\right)}{2} \sqrt{2}=\frac{1-\sqrt{2}}{2}

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