Question

Determine o conjunto solução das seguintes equações biquadradas: x4-5x2+4=0

Answer 1

Podemos Associar y = x²

y² - 5y + 4 = 0
∆= 25 - 16
∆= 9

y' = 5 + 3/2 = 4
y" = 5-3/2 = 1


x² = y'
x= √1
x = ±1

x² = y"
x = √4
x = ±2

S = { 1,-1,2,-2}

Answer 2

$\mathsf{ x^4-5x^2+4=0}$

$\mathsf{ a=1\quad b=-5\quad c=4}$

$\mathsf{\Delta=b^2-4\cdot a\cdot c }$

$\mathsf{\Delta=(-5)^2-4\cdot 1\cdot4 }$

$\mathsf{ \Delta=25-16 }$

$\mathsf{\Delta=9 }$

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

$\mathsf{x=\pm\sqrt{\dfrac{-(-5)\pm\sqrt{9}}{2\cdot 1} }}$

$\mathsf{ x=\pm\sqrt{\dfrac{5\pm3}{2}}\begin{cases}\sf x_1=+\sqrt{\dfrac{5+3}{2}}=+\sqrt{4}=+2\\\\\sf x_2=-\sqrt{\dfrac{5+3}{2}}=-\sqrt4=-2\\\\\sf x_3=+\sqrt{\dfrac{5-3}{2}}=+\sqrt1=+1\\\\\sf x_4=-\sqrt{\dfrac{5-3}{2}}=-\sqrt1=-1\end{cases}}$

$\boxed{\boxed{ \mathsf{ S=\{-2;~-1;~+1;~+2\}}}}$