Rozwiąż równania ; a) -2x^4 + 12x^3-6x^2+36x=0 b) -3x^4+21x^3-6x^2+42x=0 c)x^4+4x^3-27x-108=0 d) x^4.... Question from @marzi1267

October 2018 1 42 Report

Rozwiąż równania ; a) -2x^4 + 12x^3-6x^2+36x=0 b) -3x^4+21x^3-6x^2+42x=0 c)x^4+4x^3-27x-108=0 d) x^4-3x^3+64x-192=0 e) x^4-5x^3-8x+40=0 f) x^4+2x^3-125x-250=0

Roma

$-2x^4 + 12x^3-6x^2+36x=0 \\\ -2x \cdot (x^3 - 6x^2 + 3x - 18) = 0 \ /:(-2) \\\ x \cdot [x^2 \cdot (x - 6) + 3 \cdot (x - 6)] = 0 \\\ x \cdot (x - 6)(x^2 + 3) = 0 \\\ x = 0 \ lub \ x - 6 = 0 \ lub \ x^2 + 3 = 0 \\\\ x = 0 \\\\ x - 6 = 0 \\\ x = 6 \\\\ x^2 + 3 = 0 \\\ x^2 = - 3 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{0; \ 6 \}$

Odp. x = 0 i x = 6

$-3x^4+21x^3-6x^2+42x=0 \\\ -3x \cdot (x^3 - 7x^2 + 2x - 14) = 0 \ /:(-3) \\\ x \cdot [x^2 \cdot(x - 7) + 2 \cdot (x - 7)] = 0 \\\ x \cdot (x -7)(x^2 + 2) = 0 \\\ x = 0 \ lub \ x - 7 = 0 \ lub \ x^2 + 2 = 0 \\\\ x = 0 \\\\ x - 7 = 0 \\\ x = 7 \\\\ x^2 + 2 = 0 \\\ x^2 = - 2 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{0; \ 7 \}$

Odp. x = 0 i x = 7

$x^4+4x^3-27x-108=0 \\\ x^3 \cdot (x + 4) - 27 \cdot (x + 4) = 0 \\\ (x + 4)(x^3 - 27) = 0 \\\ (x + 4)(x^3 - 3^3) = 0 \\\ (x + 4)(x - 3)(x^2 + 3x + 9) = 0 \\\ x + 4 = 0 \ lub \ x - 3 = 0 \ x^2 + 3x + 9 = 0 \\\\ x + 4 = 0 \\\ x = - 4 \\\\ x - 3 = 0 \\\ x = 3 \\\\ x^2 + 3x + 9 = 0 \\\ \Delta = 3^2 - 4 \cdot 1 \cdot 9 = 9 - 36 = - 27 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{-4; \ 3 \}$

Odp. x = - 4 i x = 3

$x^4-3x^3+64x-192=0 \\\ x^3 \cdot (x - 3)+ 64 \cdot (x - 3) = 0 \\\ (x - 3)(x^3 + 64) = 0 \\\ (x - 3)(x^3 + 4^3) = 0 \\\ (x - 3)(x + 4)(x^2 - 4x + 16) = 0 \\\ x - 3 = 0 \ lub \ x + 4 = 0 \ lub \ x^2 - 4x + 16 = 0 \\\\ x - 3 = 0 \\\ x = 3 \\\\ x + 4 = 0 \\\ x = - 4 \\\\ x^2 - 4x + 16 = 0 \\\ \Delta = (-4)^2 - 4 \cdot 1 \cdot 16 = 16 - 64 = - 48 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{-4; \ 3 \}$

Odp. x = - 4 i x = 3

$x^4-5x^3-8x+40=0 \\\ x^3 \cdot (x - 5) - 8 \cdot (x - 5) = 0 \\\ (x - 5)(x^3 - 8) = 0 \\\ (x - 5)(x^3 - 2^3) = 0 \\\ (x - 5)(x - 2)(x^2 + 2x + 4) = 0 \\\ x - 5 = 0 \ lub \ x - 2 = 0 \ lub \ x^2 + 2x + 4 = 0 \\\\ x - 5 = 0 \\\ x = 5 \\\\ x - 2 = 0 \\\ x = 2 \\\\ x^2 + 2x + 4 = 0 \\\ \Delta = 2^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = - 8 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{2; \ 5 \}$

Odp. x = 2 i x = 5

$x^4+2x^3-125x-250=0 \\\ x^3 \cdot (x + 2) - 125 \cdot (x + 2) = 0 \\\ (x + 2)(x^3 - 125) = 0 \\\ (x + 2)(x^3 - 5^3) = 0 \\\ (x + 2)(x - 5)(x^2 + 5x + 25) = 0 \\\ x + 2 = 0 \ lub \ x - 5 = 0 \ lub \ x^2 + 5x + 25 = 0 \\\\ x + 2 = 0 \\\ x = - 2 \\\\ x - 5 = 0 \\\ x = 5 \\\\ x^2 + 5x + 25 = 0 \\\ \Delta = 5^2 - 4 \cdot 1 \cdot 25 = 25 - 100 = - 75 < 0 \\\ r\'ownanie \ nie \ ma \ rozwiaza\'n \\\\ x \in \{-2; \ 5 \}$