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Question Number 47331 by MJS last updated on 08/Nov/18

$$\mathrm{this}\mathrm{remained}\mathrm{unsolved}...$$$$\mid x-\frac{3}{4}\mid \times \mid x+\frac{5}{4}\mid =3;x\in \mathbb{C}$$

Commented by MJS last updated on 08/Nov/18

$$\mid a-\frac{3}{4}+b\mathrm{i}\mid \times \mid a+\frac{5}{4}+b\mathrm{i}\mid =3\mathrm{with}a,b\in \mathbb{R}$$$$\sqrt{{(a-\frac{3}{4})}^2+b^2}\sqrt{{(a+\frac{5}{4})}^2+b^2}=3$$$$b^4+(2a^2+a+\frac{17}{8})b^2+(a^4+a^3-\frac{13}{8}{a}^2-\frac{15}{16}a-\frac{2079}{256})=0$$$$b^2=-a^2-\frac{a}{2}-\frac{17}{16}\pm \sqrt{4a^2+2a+\frac{37}{4}}$$$$b\in \mathbb{R}\Rightarrow b=\pm \sqrt{-a^2-\frac{a}{2}-\frac{17}{16}+\sqrt{4a^2+2a+\frac{37}{4}}}$$$$\sqrt{4a^2+2a+\frac{37}{4}}⩾a^2+\frac{a}{2}+\frac{17}{16}\Rightarrow -\frac{9}{4}⩽a⩽\frac{7}{4}$$$$$$$$\mathrm{solution}\mathrm{is}$$$$x=a\pm \mathrm{i}\sqrt{-a^2-\frac{a}{2}-\frac{17}{16}+\sqrt{4a^2+2a+\frac{37}{4}}}\mathrm{with}-\frac{9}{4}⩽a⩽\frac{7}{4}$$

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