Biquadratic

By introducing a new variable solve biquadratic equation:

x4+277x215876=0 - x 4 +277 x2 -15876=0


Result

x1 =  9
x2 =  -9
x3 =  14
x4 =  -14

Solution:

y=x2  y2+277y15876=0 y2277y+15876=0  a=1;b=277;c=15876 D=b24ac=27724115876=13225 D>0  y1,2=b±D2a=277±132252 y1,2=277±1152 y1,2=138.5±57.5 y1=196 y2=81   Factored form of the equation:  (y196)(y81)=0   x1=+y2=9 x2=y2=9 x3=+y1=14 x4=y1=14y=x^2 \ \\ \ \\ -y^2 +277y -15876 =0 \ \\ y^2 -277y +15876 =0 \ \\ \ \\ a=1; b=-277; c=15876 \ \\ D = b^2 - 4ac = 277^2 - 4\cdot 1 \cdot 15876 = 13225 \ \\ D>0 \ \\ \ \\ y_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 277 \pm \sqrt{ 13225 } }{ 2 } \ \\ y_{1,2} = \dfrac{ 277 \pm 115 }{ 2 } \ \\ y_{1,2} = 138.5 \pm 57.5 \ \\ y_{1} = 196 \ \\ y_{2} = 81 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (y -196) (y -81) = 0 \ \\ \ \\ \ \\ x_1 = +\sqrt{y_2} = 9 \ \\ x_2 = -\sqrt{y_2} = -9 \ \\ x_3 = +\sqrt{y_1} = 14 \ \\ x_4 = -\sqrt{y_1} = -14



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