Question

Resolva e C a equação: x⁴ + 3x³ - 2x² + 3x + 1 = 0

Answer 1

Resposta:

Explicação passo-a-passo:

x⁴ + 3x³ - 2x² + 3x + 1 = 0

Dividindo por x²

$x^{2} +3x-2+\frac{3}{x}+\frac{1}{x^{2}} =0\\\\x^{2}+\frac{1}{x^{2}}+3x+\frac{3}{x}-2=0\\\\\ x^{2} +\frac{1}{x^{2}}+3(x+\frac{1}{x})-2=0$

$Fazendo\:x+\frac{1}{x}=y\\\\Quadrando\\\\x^{2}+2x\frac{1}{x}+\frac{1}{x^{2}}=y^{2}\\\\x^{2}+\frac{1}{x^{2}}=y^{2}-2\\\\Subst.\:fica\\\\y^{2}-2+3y-2=0\\\\y^{2}+3y-4=0$}

Δ = 3² - 4.1.(-4)

Δ = 9 + 16

Δ = 25

$y=\frac{-3-5}{2}=\frac{-8}{2}=-4\\ ou\\y=\frac{-3+5}{2}=\frac{2}{2}=1\\\\x+\frac{1}{x}=-4\\\\x^{2}+4x+1=0\\\\[tex]Delta=4^{2}-4.1.4=12\\\\x=\frac{-4-2\sqrt{3}}{2}=-2-\sqrt{3}\\ou\\x=\frac{-4+2\sqrt{3}}{2}=-2+\sqrt{3}\\\\\\[tex]x+\frac{1}{x}=1\\\\x^{2}-x+1=0\\\\Delta=1-4=-3\\\\x=\frac{1-\sqrt{-3}}{2}=\frac{1}{2}-\frac{3i}{2}\\ou\\x=\frac{1}{2}+\frac{3i}{2}$

S = { $-2-\sqrt{3},-2+\sqrt{3},\frac{1}{2}-\frac{3i}{2},\frac{1}{2}+\frac{3i}{2}$}