Question

Rozwiąż równanie

x^3-5x^2+5x+2=0

Answer 1

Odpowiedź:

[tex]\huge\boxed{x = \frac{3-\sqrt{13}}{2} \ \vee \ x = 2 \ \vee \ x = \frac{3+\sqrt{13}}{2}}[/tex]

Szczegółowe wyjaśnienie:

[tex]x^{3}-5x^{2}{+5x+2=0[/tex]

[tex]x^{3}-3x^{2}-x - 2x^{2}+6x+2 = 0[/tex]

[tex]x(x^{2}-3x-1)-2(x^{2}-3x-1) = 0[/tex]

[tex](x-2)(x^{2}-3x-1) = 0[/tex]

[tex]x-2 = 0 \ \vee \ x^{2}-3x-1 = 0[/tex]

1.

[tex]x-2 = 0\\\\\boxed{x = 2}[/tex]

Lub

2.

[tex]x^{2}-3x-1 = 0\\\\a = 1, \ b = -3, \ c = -1\\\\\Delta = b^{2}-4ac = (-3)^{2}-4\cdot1\cdot(-1) =9+4 = 13\\\\\sqrt{\Delta} = \sqrt{13}\\\\x_1 = \frac{-b-\sqrt{\Delta}}{2a} = \frac{-(-3)-\sqrt{13}}{2\cdot1} =\boxed{\frac{3-\sqrt{13}}{2}}\\\\x_2 =\frac{-b+\sqrt{\Delta}}{2a} = \frac{-(-3)+\sqrt{13}}{2\cdot 1} = \boxed{\frac{3+\sqrt{13}}{2}}[/tex]