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x^4+2x^2-3=0

x^4+2x^2-3=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 4      2        
x  + 2*x  - 3 = 0
x4+2x2−3=0x^{4} + 2 x^{2} - 3 = 0
Simplify
Detail solution
Given the equation:
x4+2x2−3=0x^{4} + 2 x^{2} - 3 = 0
Do replacement
v=x2v = x^{2}
then the equation will be the:
v2+2v−3=0v^{2} + 2 v - 3 = 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=−3c = -3
, then
D=b2−4∗a∗c=D = b^2 - 4 * a * c =
22−1⋅4(−3)=162^{2} - 1 \cdot 4 \left(-3\right) = 16
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=1v_{1} = 1
Simplify
v2=−3v_{2} = -3
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⋅1121=1x_{1} = \frac{0}{1} + \frac{1 \cdot 1^{\frac{1}{2}}}{1} = 1
x2=(−1)1121+01=−1x_{2} = \frac{\left(-1\right) 1^{\frac{1}{2}}}{1} + \frac{0}{1} = -1
x3=01+1(−3)121=3ix_{3} = \frac{0}{1} + \frac{1 \left(-3\right)^{\frac{1}{2}}}{1} = \sqrt{3} i
x4=01+(−1)(−3)121=−3ix_{4} = \frac{0}{1} + \frac{\left(-1\right) \left(-3\right)^{\frac{1}{2}}}{1} = - \sqrt{3} i
The graph
05-15-10-51015-100100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x_1 = -1
x1=−1x_{1} = -1
Simplify 
x_2 = 1
x2=1x_{2} = 1
Simplify 
           ___
x_3 = -I*\/ 3
x3=−3ix_{3} = - \sqrt{3} i
Simplify 
          ___
x_4 = I*\/ 3
x4=3ix_{4} = \sqrt{3} i
Simplify
Sum and product of roots [src] 
sum
              ___       ___
-1 + 1 + -I*\/ 3  + I*\/ 3
(−1)+(1)+(−3i)+(3i)\left(-1\right) + \left(1\right) + \left(- \sqrt{3} i\right) + \left(\sqrt{3} i\right) 
=
0
00
Simplify 
product
              ___       ___
-1 * 1 * -I*\/ 3  * I*\/ 3
(−1)∗(1)∗(−3i)∗(3i)\left(-1\right) * \left(1\right) * \left(- \sqrt{3} i\right) * \left(\sqrt{3} i\right) 
=
-3
−3-3
Simplify
Numerical answer [src] 
x1 = 1.0
x2 = 1.73205080756888*i
x3 = -1.73205080756888*i
x4 = -1.0
x4 = -1.0
The graph
x^4+2x^2-3=0 equation
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