Question

Resolva as equações em C.
a) x2 + 81 = 0
b) x2 - 4x + 5 = 0
c) - x2 - 2x - 2 = 0
d) x2 + 3 = 0​

Answer 1

Resolver as equações em C, ou seja, no conjunto dos complexos

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Lembre-se que uma unidade imaginária elevada ao quadrado é igual a menos um

$\boxed{i^2=-1}$

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$\underbrace{\sf Veja:}$

a)

$\sf x^2+81=0$

$\sf x^2=-81$

$\sf x=\pm~\sqrt{-81}~~\Rightarrow~~x \notin \mathbb{R}$

$\sf x=\pm~\sqrt{81.(-1)}$

$\sf x=\pm~(\sqrt{81}\cdot\sqrt{-1})$

$\sf x=\pm~(\sqrt{81}\cdot\sqrt{i^2})$

$\sf x=\pm~(9\cdot i)$

$\sf x=\pm~9i$

o conjunto solução é:

$\boxed{\sf S=\left\{-9i~~;~~9i\right\}}$

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b)

$\sf x^2-4x+5=0$

coeficientes: a = 1, b = -4, c = 5

$\sf \Delta=b^2-4ac$

$\sf \Delta=(-4)^2-4.(1).(5)$

$\sf \Delta=16-20$

$\sf \Delta=-4~~\Rightarrow~~x \notin \mathbb{R}$

$\sf \Delta=4.(-1)$

$\sf \Delta=4.(i^2)$

$\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}$

$\sf x=\dfrac{-(-4)\pm\sqrt{4.(i^2)}}{2.(1)}$

$\sf x=\dfrac{4\pm\sqrt{4}\cdot\sqrt{I^2}}{2}$

$\sf x=\dfrac{4\pm2\cdot i}{2}~\Rightarrow~x=\dfrac{4\pm2i}{2}$

$\sf x'=\dfrac{4+2i}{2}~\Rightarrow~\boxed{\sf x'=2+i}$

$\sf x''=\dfrac{4-2i}{2}~\Rightarrow~\boxed{\sf x''=2-i}$

o conjunto solução é:

$\boxed{\sf S=\left\{2+i~;~2-i\right\}}$

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c)

$\sf -x^2-2x-2=0$

coeficientes: a = -1, b = -2, c = -2

$\sf \Delta=b^2-4ac$

$\sf \Delta=(-2)^2-4.(-1).(-2)$

$\sf \Delta=4-8$

$\sf \Delta=-4~~\Rightarrow~~x \notin \mathbb{R}$

$\sf \Delta=4.(-1)$

$\sf \Delta=4.(i^2)$

$\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}$

$\sf x=\dfrac{-(-2)\pm\sqrt{4.(i^2)}}{2.(-1)}$

$\sf x=\dfrac{2\pm\sqrt{4}\cdot\sqrt{I^2}}{-2}$

$\sf x=\dfrac{2\pm2\cdot i}{-2}~\Rightarrow~x=\dfrac{4\pm2i}{-2}$

$\sf x'=\dfrac{2+2i}{-2}~\Rightarrow~\boxed{\sf x'=-1-i}$

$\sf x''=\dfrac{2-2i}{-2}~\Rightarrow~\boxed{\sf x''=-1+i}$

o conjunto solução é:

$\boxed{\sf S=\left\{-1-i~;~-1+i\right\}}$

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d)

$\sf x^2+3=0$

$\sf x^2=-3$

$\sf x=\pm~\sqrt{-3}~~\Rightarrow~~x \notin \mathbb{R}$

$\sf x=\pm~\sqrt{3.(-1)}$

$\sf x=\pm~(\sqrt{3}\cdot\sqrt{-1})$

$\sf x=\pm~(\sqrt{3}\cdot\sqrt{i^2})$

$\sf x=\pm~(\sqrt{3}\cdot i)$

$\sf x=\pm~\sqrt{3}~i$

o conjunto solução é:

$\boxed{\sf S=\left\{-\sqrt{3}~i~~;~~\sqrt{3}~i\right\}}$