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Roots of biquadratic equations


If you want to look for the roots of a biquadratic equation, there are three ways available in Orimath Biquadratic Solver. In the problem presented to you, you might be asked to use either quadratic formula, factorization or completing the square. Orimath Biquadratic Solver can show you how to use any of them to solve a biquadratic polynomial equation.


All you have to do is to check any check boxes related to solving biquadratic equation ( see the picture above ). Then click as HTML to tell Orimath Biquadratic Solver to solve the problem for you and tell you the steps required to get the answer.




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Answer by Orimath Quadratic Solver :
Question Number 1 :
For this equation ( x2 + 3 ) ( x2 - 2 ) = x2 - 2 , answer the following questions :
A.
Find the roots using Quadratic Formula ! !
B.
Use completing the square to find the root of the equation !

Answer Number 1 :
First, we must turn this equation ( x2 + 3 ) ( x2 - 2 ) = x^2 - 2 into ax2+bx+c=0 form.
( x2 + 3 ) ( x2 - 2 ) = x2 - 2 , expand the left hand side.
<=> x2 ( x2 - 2 ) + 3 ( x2 - 2 ) = x2 - 2
<=> ( x4 - 2x2 ) + ( 3x2 - 6 ) = x2 - 2
<=> x4 + x2 - 6 = x2 - 2 , move everything in the right hand side to the left hand side.
<=> x4 + x2 - 6 - ( x2 - 2 ) = 0
<=> x4 + x2 - 6 + ( - x2 + 2 ) = 0
<=> x4 - 4 = 0
The equation x4 - 4 = 0 is already in ax4+bx2+c=0 form.
In that form, we can easily derive that the value of a = 1, b = 0, c = -4.

By using abc formula the value of x as defined by
x2[1,2] =
-b (b2-4ac)
2a
As a = 1, b = 0 and c = -4,
we need to subtitute a,b,c in the abc formula, with those values.
So x2[1,2] = - (0) (0)2 - 4 (1) (-4) )
2 (1)
So x2[1,2] = 0(0 + 16)
2
So x2[1,2] = 016
2
We can get x12 = ( 4 )/(2) and x22 = ( - 4 )/(2)
We can get x12 = 2 and x22 = -2
The equation x4 - 4 = 0 , have four roots :
Root 1 : x1 =      x12   = 2   = 1.4142135623731
Root 2 : x2 =      x22   = -2   = 1.4142135623731i
Root 3 : x3 = -     x12   = -2   = -1.4142135623731
Root 4 : x4 = -     x22   = --2   = -1.4142135623731i
x4 - 4 = 0 ,divide both side with 1
So we get x4 - 4 = 0 ,
The coefficient of x is 0
We have to use the fact that ( x + q )2 = x2 + 2qx + q2 , assume that q = 0/2 = 0 , and q2 = (0)2 = 0
Which means we can turn the equation into x4 - 4 = 0
So we will get ( x2 )2 - 4 = 0
And it is the same with (( x2 ) - 2 ) (( x2 ) + 2 ) = 0
By using the associative law we get ( x2 - 2 ) ( x2 + 2 ) = 0
Just add up the constants in each brackets, and we get ( x2 - 2 ) ( x2 + 2 ) = 0
We got x12 = 2 and x22 = -2
The equation x4 - 4 = 0 , have four roots :
Root 1 : x1 =      x12   = 2   = 1.4142135623731
Root 2 : x2 =      x22   = -2   = 1.4142135623731i
Root 3 : x3 = -     x12   = -2   = -1.4142135623731
Root 4 : x4 = -     x22   = --2   = -1.4142135623731i

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