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x^4-2x^2-8=0

x^4-2x^2-8=0 equation

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Numerical solution:

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The solution

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 4      2        
x  - 2*x  - 8 = 0
x4−2x2−8=0x^{4} - 2 x^{2} - 8 = 0
Simplify
Detail solution
Given the equation:
x4−2x2−8=0x^{4} - 2 x^{2} - 8 = 0
Do replacement
v=x2v = x^{2}
then the equation will be the:
v2−2v−8=0v^{2} - 2 v - 8 = 0
This equation is of the form
a v2+b v+c=0a\ v^2 + b\ v + c = 0
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
v1=D−b2av_{1} = \frac{\sqrt{D} - b}{2 a}
v2=−D−b2av_{2} = \frac{- \sqrt{D} - b}{2 a}
where D=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=1a = 1
b=−2b = -2
c=−8c = -8
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
(−2)2−1⋅4(−8)=36\left(-2\right)^{2} - 1 \cdot 4 \left(-8\right) = 36
Because D > 0, then the equation has two roots.
v1=(−b+D)2av_1 = \frac{(-b + \sqrt{D})}{2 a}
v2=(−b−D)2av_2 = \frac{(-b - \sqrt{D})}{2 a}
or
v1=4v_{1} = 4
Simplify
v2=−2v_{2} = -2
Simplify
The final answer:
Because
v=x2v = x^{2}
then
x1=v1x_{1} = \sqrt{v_{1}}
x2=−v1x_{2} = - \sqrt{v_{1}}
x3=v2x_{3} = \sqrt{v_{2}}
x4=−v2x_{4} = - \sqrt{v_{2}}
then:
x1=01+1⋅4121=2x_{1} = \frac{0}{1} + \frac{1 \cdot 4^{\frac{1}{2}}}{1} = 2
x2=(−1)4121+01=−2x_{2} = \frac{\left(-1\right) 4^{\frac{1}{2}}}{1} + \frac{0}{1} = -2
x3=01+1(−2)121=2ix_{3} = \frac{0}{1} + \frac{1 \left(-2\right)^{\frac{1}{2}}}{1} = \sqrt{2} i
x4=01+(−1)(−2)121=−2ix_{4} = \frac{0}{1} + \frac{\left(-1\right) \left(-2\right)^{\frac{1}{2}}}{1} = - \sqrt{2} i
The graph
02468-21810121416-100100
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
              ___       ___
-2 + 2 + -I*\/ 2  + I*\/ 2
(−2)+(2)+(−2i)+(2i)\left(-2\right) + \left(2\right) + \left(- \sqrt{2} i\right) + \left(\sqrt{2} i\right) 
=
0
00
Simplify 
product
              ___       ___
-2 * 2 * -I*\/ 2  * I*\/ 2
(−2)∗(2)∗(−2i)∗(2i)\left(-2\right) * \left(2\right) * \left(- \sqrt{2} i\right) * \left(\sqrt{2} i\right) 
=
-8
−8-8
Simplify
Rapid solution [src] 
x_1 = -2
x1=−2x_{1} = -2
Simplify 
x_2 = 2
x2=2x_{2} = 2
Simplify 
           ___
x_3 = -I*\/ 2
x3=−2ix_{3} = - \sqrt{2} i
Simplify 
          ___
x_4 = I*\/ 2
x4=2ix_{4} = \sqrt{2} i
Simplify
Numerical answer [src] 
x1 = -1.4142135623731*i
x2 = 2.0
x3 = 1.4142135623731*i
x4 = -2.0
x4 = -2.0
The graph
x^4-2x^2-8=0 equation
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