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2x^4-3x^2=0

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

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Numerical solution:

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The solution

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   4      2    
2*x  - 3*x  = 0
2x4−3x2=02 x^{4} - 3 x^{2} = 0
Simplify
Detail solution
Given the equation:
2x4−3x2=02 x^{4} - 3 x^{2} = 0
Do replacement
v=x2v = x^{2}
then the equation will be the:
2v2−3v=02 v^{2} - 3 v = 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=2a = 2
b=−3b = -3
c=0c = 0
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
(−1)2⋅4⋅0+(−3)2=9\left(-1\right) 2 \cdot 4 \cdot 0 + \left(-3\right)^{2} = 9
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=32v_{1} = \frac{3}{2}
Simplify
v2=0v_{2} = 0
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(32)121=62x_{1} = \frac{0}{1} + \frac{1 \left(\frac{3}{2}\right)^{\frac{1}{2}}}{1} = \frac{\sqrt{6}}{2}
x2=(−1)(32)121+01=−62x_{2} = \frac{\left(-1\right) \left(\frac{3}{2}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \frac{\sqrt{6}}{2}
x3=1⋅0121+01=0x_{3} = \frac{1 \cdot 0^{\frac{1}{2}}}{1} + \frac{0}{1} = 0
The graph
05-15-10-51015-100100
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
       ___      ___
    -\/ 6     \/ 6 
0 + ------- + -----
       2        2
(0)+(−62)+(62)\left(0\right) + \left(- \frac{\sqrt{6}}{2}\right) + \left(\frac{\sqrt{6}}{2}\right) 
=
0
00
Simplify 
product
       ___      ___
    -\/ 6     \/ 6 
0 * ------- * -----
       2        2
(0)∗(−62)∗(62)\left(0\right) * \left(- \frac{\sqrt{6}}{2}\right) * \left(\frac{\sqrt{6}}{2}\right) 
=
0
00
Simplify
Rapid solution [src] 
x_1 = 0
x1=0x_{1} = 0
Simplify 
         ___ 
      -\/ 6  
x_2 = -------
         2
x2=−62x_{2} = - \frac{\sqrt{6}}{2}
Simplify 
        ___
      \/ 6 
x_3 = -----
        2
x3=62x_{3} = \frac{\sqrt{6}}{2}
Simplify
Numerical answer [src] 
x1 = 0.0
x2 = -1.22474487139159
x3 = 1.22474487139159
x3 = 1.22474487139159
The graph
2x^4-3x^2=0 equation
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