Answer on Question #44538 - Math - Other
a) Solve the equation 9 x 4 − 18 x 3 − 31 x 2 + 8 x + 12 = 0 9x^4 - 18x^3 - 31x^2 + 8x + 12 = 0 9 x 4 − 18 x 3 − 31 x 2 + 8 x + 12 = 0
We have an equation a x 4 + b x 3 + c x 2 + d x + e = 0 ax^4 + bx^3 + cx^2 + dx + e = 0 a x 4 + b x 3 + c x 2 + d x + e = 0
where a = 9 a = 9 a = 9 b = − 18 b = -18 b = − 18 c = − 31 c = -31 c = − 31 d = 8 d = 8 d = 8 e = 12 e = 12 e = 12
a l p h a = c a − 3 b 2 8 a 2 = − 4.944444 alpha = \frac{c}{a} - \frac{3b^2}{8a^2} = -4.944444 a lp ha = a c − 8 a 2 3 b 2 = − 4.944444 b e t a = b 3 8 a 3 − b c 2 a 2 + d a = − 3.555556 beta = \frac{b^3}{8a^3} - \frac{bc}{2a^2} + \frac{d}{a} = -3.555556 b e t a = 8 a 3 b 3 − 2 a 2 b c + a d = − 3.555556 g a m m a = − 3 b 4 256 a 4 + c b 2 16 a 3 − b d 4 a 2 + e a = 0.7291667 gamma = \frac{-3b^4}{256a^4} + \frac{cb^2}{16a^3} - \frac{bd}{4a^2} + \frac{e}{a} = 0.7291667 g amma = 256 a 4 − 3 b 4 + 16 a 3 c b 2 − 4 a 2 b d + a e = 0.7291667 P = − a l p h a 2 12 − g a m m a = − 2.766461 P = \frac{-alpha^2}{12} - gamma = -2.766461 P = 12 − a lp h a 2 − g amma = − 2.766461 Q = − a l p h a 3 108 + a l p h a g a m m a 3 − b e t a 2 8 = − 1.662767 Q = \frac{-alpha^3}{108} + alpha \frac{gamma}{3} - \frac{beta^2}{8} = -1.662767 Q = 108 − a lp h a 3 + a lp ha 3 g amma − 8 b e t a 2 = − 1.662767 R p = Q 2 + Q 2 4 + P 3 27 = − 0.8313837 + 0.3049106 i R_p = \frac{Q}{2} + \sqrt{\frac{Q^2}{4} + \frac{P^3}{27}} = -0.8313837 + 0.3049106i R p = 2 Q + 4 Q 2 + 27 P 3 = − 0.8313837 + 0.3049106 i R m = Q 2 − Q 2 4 + P 3 27 = − 0.8313837 − 0.3049106 i R_m = \frac{Q}{2} - \sqrt{\frac{Q^2}{4} + \frac{P^3}{27}} = -0.8313837 - 0.3049106i R m = 2 Q − 4 Q 2 + 27 P 3 = − 0.8313837 − 0.3049106 i U = R m ( 1 3 ) = 0.5740741 − 0.7698004 i U = R_m^{\left(\frac{1}{3}\right)} = 0.5740741 - 0.7698004i U = R m ( 3 1 ) = 0.5740741 − 0.7698004 i y = − 5 a l p h a 6 − U + P 3 U = 2.972222 y = -5 \frac{alpha}{6} - U + \frac{P}{3U} = 2.972222 y = − 5 6 a lp ha − U + 3 U P = 2.972222 W = a l p h a + 2 y = 1 W = \sqrt{alpha + 2y} = 1 W = a lp ha + 2 y = 1
The roots are:
x 1 = − b 4 a + W − 3 a l p h a + 2 y + 2 b e t a W 2 x_1 = \frac{-b}{4a} + \frac{W - \sqrt{3alpha + 2y + 2 \frac{beta}{W}}}{2} x 1 = 4 a − b + 2 W − 3 a lp ha + 2 y + 2 W b e t a x 2 = − b 4 a + − W + 3 a l p h a + 2 y + 2 b e t a W 2 x_2 = \frac{-b}{4a} + \frac{-W + \sqrt{3alpha + 2y + 2 \frac{beta}{W}}}{2} x 2 = 4 a − b + 2 − W + 3 a lp ha + 2 y + 2 W b e t a x 3 = − b 4 a + W − 3 a l p h a + 2 y + 2 b e t a W 2 x_3 = \frac{-b}{4a} + \frac{W - \sqrt{3alpha + 2y + 2 \frac{beta}{W}}}{2} x 3 = 4 a − b + 2 W − 3 a lp ha + 2 y + 2 W b e t a x 4 = − b 4 a + − W − 3 a l p h a + 2 y + 2 b e t a W 2 x_4 = \frac{-b}{4a} + \frac{-W - \sqrt{3alpha + 2y + 2 \frac{beta}{W}}}{2} x 4 = 4 a − b + 2 − W − 3 a lp ha + 2 y + 2 W b e t a x 1 = 3 , x 2 = 2 3 , x 3 = − 1 , x 4 = − 2 3 x_1 = 3, x_2 = \frac{2}{3}, x_3 = -1, x_4 = -\frac{2}{3} x 1 = 3 , x 2 = 3 2 , x 3 = − 1 , x 4 = − 3 2
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#340153 on Dec 2023