Question

01) Resolva as equações do 2º grau abaixo, dentro do conjunto

dos números complexos: ℂ

a) x

2 + 4 = 0

b) x

2 + 25 = 0

c) x

2 – 2x + 5 = 0

d) x

2 – 4x + 5 = 0

e) x

2 + 36x = 0​

Answer 1

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$\tt{a)}~\sf{x^2+4=0}\\\sf{x^2=-4}\\\sf{x=\pm\sqrt{-4}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{x=\pm2~i}}}}}\\\tt{b)}~\sf{x^2+25=0}\\\sf{x^2=-25}}\\\sf{x=\pm\sqrt{25}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{x=\pm5i}}}}}\\\tt{c)}~\sf{x^2-2x+5=0}\\\sf{\Delta=b^2-4ac}\\\sf{\Delta=(-2)^2-4\cdot1\cdot5}\\\sf{\Delta=4-20}\\\sf{\Delta=-16}\\\sf{x=\dfrac{-b\pm\sqrt{\Delta}}{2a}}\\\sf{x=\dfrac{-(-2)\pm\sqrt{-16}}{2\cdot1}}\\\sf{x=\dfrac{2\pm4i}{2}=\dfrac{\diagdown\!\!\!\!2\cdot(1\pm2i)}{\diagdown\!\!\!\!2}}$

$\sf{x=1\pm2i}\begin{cases}\sf{x_1=1+2i}\\\sf{x_2=1-2i}\end{cases}$

$\tt{d)}~\sf{x^2-4x+5=0}\\\sf{\Delta=b^2-4ac}\\\sf{\Delta=(-4)^2-4\cdot1\cdot5}\\\sf{\Delta=16-20}\\\sf{\Delta=-4}\\\sf{x=\dfrac{-b\pm\sqrt{\Delta}}{2a}}\\\sf{x=\dfrac{-(-4)\pm\sqrt{-4}}{2\cdot1}}\\\sf{x=\dfrac{4\pm2i}{2}=\dfrac{\diagdown\!\!\!\!2\cdot(2\pm i)}{\diagdown\!\!\!\!2}}\\\sf{x=2\pm i}\begin{cases}\sf{x_1=2+i}\\\sf{x_2=2-i}\end{cases}$

$\tt{e)}~\sf{x^2+36x=0}\\\sf{x\cdot(x+36)=0}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{x=0}}}}}\\\sf{x+36=0}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{x=-36}}}}}$