Question

resolva as equações do segundo grau abaixo, obtendo as respostas na forma de um número Complexo: a) x 2 + 2x + 10 = 0 b) x 2 + 6x + 25 = 0 c) x 2 – x + 2 = 0 d) x 2 + 25 = 0

Answer 1

Explicação passo-a-passo:

a) $\sf x^2+2x+10=0$

$\sf \Delta=2^2-4\cdot1\cdot10$

$\sf \Delta=4-40$

$\sf \Delta=-36$

$\sf \Delta=36\cdot(-1)$

$\sf \Delta=36i^2$

$\sf x=\dfrac{-2\pm\sqrt{36i^2}}{2\cdot1}=\dfrac{-2\pm6i}{2}$

$\sf x'=\dfrac{-2+6i}{2}~\longrightarrow~x'=-1+3i$

$\sf x"=\dfrac{-2-6i}{2}~\longrightarrow~x"=-1-3i$

$\sf S=\{-1-3i,-1+3i\}$

b) $\sf x^2+6x+25=0$

$\sf \Delta=6^2-4\cdot1\cdot25$

$\sf \Delta=36-100$

$\sf \Delta=-64$

$\sf \Delta=64\cdot(-1)$

$\sf \Delta=64i^2$

$\sf x=\dfrac{-6\pm\sqrt{64i^2}}{2\cdot1}=\dfrac{-6\pm8i}{2}$

$\sf x'=\dfrac{-6+8i}{2}~\longrightarrow~x'=-3+4i$

$\sf x"=\dfrac{-6-8i}{2}~\longrightarrow~x"=-3-4i$

$\sf S=\{-3-4i,-3+4i\}$

c) $\sf x^2-x+2=0$

$\sf \Delta=(-1)^2-4\cdot1\cdot2$

$\sf \Delta=1-8$

$\sf \Delta=-7$

$\sf \Delta=7\cdot(-1)$

$\sf \Delta=7i^2$

$\sf x=\dfrac{-(-1)\pm\sqrt{7i^2}}{2\cdot1}=\dfrac{1\pm\sqrt{7}\cdot i}{2}$

$\sf x'=\dfrac{1+\sqrt{7}\cdot i}{2}$

$\sf x"=\dfrac{1-\sqrt{7}\cdot i}{2}$

$\sf S=\left\{\dfrac{1-\sqrt{7}\cdot i}{2},\dfrac{1+\sqrt{7}\cdot i}{2}\right\}$

d) $\sf x^2+25=0$

$\sf x^2=-25$

$\sf x^2=25\cdot(-1)$

$\sf x^2=25i^2$

$\sf x=\pm\sqrt{25i^2}$

$\sf x'=5i$

$\sf x"=-5i$

$\sf S=\{-5i,5i\}$