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Biquadrades Resol les següents equacions biquadrades: a) `x^4 - 10 x^2 + 9= 0`
`t^2 - 10 t + 9= 0` `t=(10\pmsqrt((-10)^2-4·1·9))/(2·1)` `t=(10\pmsqrt(100-36))/2` `t=(10\pmsqrt(64))/2` `t=(10\pm8)/2` `t_1 = (10+8)/2 = 18/2 = 9` `t_2 = (10-8)/2 = 2/2 = 1` `x^2 = 9 => x = sqrt(9) =\pm3` `x^2 = 1 => x = sqrt(1) =\pm1` `x_1 = 3, x_2 = -3, x_3 = 1, x_4 = -1` b) `x^4 - 20 x^2 + 64= 0`
`t^2 - 20 t + 64= 0` `t=(20\pmsqrt((-20)^2-4·64))/(2·1)` `t=(20\pmsqrt(400-256))/2` `t=(20\pmsqrt(144))/2` `t=(20\pm12)/2` `t_1 = (20+12)/2 = 32/2 = 16` `t_2 = (20-12)/2 = 8/2 = 4` `x^2 = 16 => x = sqrt(16) =\pm4` `x^2 = 4 => x = sqrt(1) =\pm2` `x_1 = 4, x_2 = -4, x_3 = 2, x_4 = -2` c) `x^4 - 7x^2 - 18= 0`
`t^2 - 7 t - 18= 0` `t=(7\pmsqrt((-7)^2-4·1·(-18)))/(2·1)` `t=(7\pmsqrt(49+72))/2` `t=(7\pmsqrt(121))/2` `t=(7\pm 11)/2` `t=(7 + 11)/2 = 9` `t=(7 - 11)/2 = -2` d'aquí no surt cap solució real Només té dues solucions `x^2 = 9 => x = sqrt(9) =\pm3` `x_1 = 3, x_2 = -3` d) `x^4 - 16x^2 = 0`
`t^2 - 16 t = 0` `t·(t - 16) = 0` `t = 0` `t - 16 = 0` `t = 16` `x^2 = 0 => x = sqrt(0) = 0` `x^2 = 16 => x = sqrt(16) =\pm4` `x_1 = 0, x_2 = 4, x_3 = -4` e) `3x^4 - 48 = 0`
`3t^2 - 48 = 0` `3t^2 = 48` `t^2 = 48/3` `t^2 = 16` `t = sqrt(16)` `t = \pm4` `x^2 = 4 => x = sqrt(4) = \pm2` `x^2 = -16 => x = sqrt(-16) = X` `x_1 = 2, x_2 = -2` f) `x^6 - 9x^3 + 8 = 0`
`t^2 - 9 t + 8= 0` `t=(9\pmsqrt(9^2-4·1·8))/(2·1)` `t=(9\pmsqrt(81-32))/2` `t=(9\pmsqrt(49))/2` `t=(9\pm7)/2` `t_1 = (9+7)/2 = 16/2 = 8` `t_2 = (9-7)/2 = 2/2 = 1` `x^3 = 8 => x =` $\sqrt[3]{8} = 2$ `x^3 = 1 => x =` $\sqrt[3]{1} = 1$ `x_1 = 1, x_2 = 2` |