User:Quantum Burrito/Complex Numbers

$$\quad (a+ai)(a-ai) =a^2-a^2i+a^2i-a^2i^2 =a^2-a^2i^2 =a^2-a^2(-1) =a^2+a^2 =2a^2 $$

$$\left(\frac{4+2i}{-1+2i}\right)^2$$

$$=\left(\frac{(4+2i)(1+2i)}{(-1+2i)(1+2i)}\right)^2$$

$$=\left(\frac{4i^2+2i+8i+4}{4i^2+2i-2i-1}\right)^2$$

$$=\left(\frac{-4+10i+4}{-4-1}\right)^2$$

$$=\left(\frac{10i}{-5}\right)^2$$

$$=\left(-2i\right)^2$$

$$\quad =-4$$

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$$\quad 5x^2+4x+1=0$$

$$5\left[x^2+\frac{4}{5}x+\frac{1}{5}\right]=0$$

$$5\left[\left(x+\frac{2}{5}\right)^2-\frac{4}{25}+\frac{1}{5}\right]=0$$

$$5\left[\left(x+\frac{2}{5}\right)^2-\frac{4}{25}+\frac{5}{25}\right]=0$$

$$5\left[\left(x+\frac{2}{5}\right)^2+\frac{1}{25}\right]=0$$

$$5\left(x+\frac{2}{5}\right)^2+\frac{5}{25}=0$$

$$5\left(x+\frac{2}{5}\right)^2=-\frac{5}{25}$$

$$\left(x+\frac{2}{5}\right)^2=-\frac{1}{25}$$

$$x+\frac{2}{5}=\pm\sqrt{-\frac{1}{25}}$$

$$x=-\frac{2}{5}\pm\sqrt{-\frac{1}{25}}$$

$$x=-\frac{2}{5}\pm\frac{1}{5}i$$

$$x=\left(-\frac{2}{5}+\frac{1}{5}i\right)\qquad OR \qquad x=\left(-\frac{2}{5}-\frac{1}{5}i\right)$$

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$$\frac{1}{(a+bi)}$$

$$=\frac{1(a-bi)}{(a+bi)(a-bi)}$$

$$=\frac{(a-bi)}{a^2-abi+abi-b^2i^2}$$

$$=\frac{(a-bi)}{a^2-b^2(-1)}$$

$$=\frac{(a-bi)}{a^2+b^2}$$