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x⁴-5x²+4=0

x⁴-5x²+4=0 equation

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Numerical solution:

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The solution

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 4      2        
x  - 5*x  + 4 = 0
x4−5x2+4=0x^{4} - 5 x^{2} + 4 = 0
Simplify
Detail solution
Given the equation:
x4−5x2+4=0x^{4} - 5 x^{2} + 4 = 0
Do replacement
v=x2v = x^{2}
then the equation will be the:
v2−5v+4=0v^{2} - 5 v + 4 = 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=−5b = -5
c=4c = 4
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
(−1)1⋅4⋅4+(−5)2=9\left(-1\right) 1 \cdot 4 \cdot 4 + \left(-5\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=4v_{1} = 4
Simplify
v2=1v_{2} = 1
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⋅1121=1x_{3} = \frac{0}{1} + \frac{1 \cdot 1^{\frac{1}{2}}}{1} = 1
x4=(−1)1121+01=−1x_{4} = \frac{\left(-1\right) 1^{\frac{1}{2}}}{1} + \frac{0}{1} = -1
The graph
05-15-10-51015-100100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x_1 = -2
x1=−2x_{1} = -2
Simplify 
x_2 = -1
x2=−1x_{2} = -1
Simplify 
x_3 = 1
x3=1x_{3} = 1
Simplify 
x_4 = 2
x4=2x_{4} = 2
Simplify
Sum and product of roots [src] 
sum
-2 + -1 + 1 + 2
(−2)+(−1)+(1)+(2)\left(-2\right) + \left(-1\right) + \left(1\right) + \left(2\right) 
=
0
00
Simplify 
product
-2 * -1 * 1 * 2
(−2)∗(−1)∗(1)∗(2)\left(-2\right) * \left(-1\right) * \left(1\right) * \left(2\right) 
=
4
44
Simplify
Numerical answer [src] 
x1 = 2.0
x2 = -1.0
x3 = -2.0
x4 = 1.0
x4 = 1.0
The graph
x⁴-5x²+4=0 equation
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