SOLUTION: Please help me solve this equation: {{{ 3*x^-4-21*x^-2+3=0 }}}

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Question 263737: Please help me solve this equation: +3%2Ax%5E-4-21%2Ax%5E-2%2B3=0+
Found 2 solutions by mananth, jsmallt9:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me solve this equation: +3%2Ax%5E-4-21%2Ax%5E-2%2B3=0+
3*x^-4-21*x^-2+3=0 multiply this equation by x^4
3_21x^2+3x^4=0
3x^4-21x^2+3=0 common factor 3
x^4-7x^2+1=0
x^2(x^2-7)+ 1 =0 Divide by x^2
x^2-7 +1/x^2=0
x^2+1/x^2-7=0
x^2+2+1/x^2-9=0
(x+1/x)^2-9=0
(x+(1/x)+3)(x+(1/x)-3)=0

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Probably the simplest solution to this equation is based on recognizing that x%5E%28-4%29+=+%28x%5E%28-2%29%29%5E2. That makes this equation quadratic in x%5E%28-2%29. It may help to use a temporary variable:
Let q+=+x%5E%28-2%29
which makes q%5E2+=+x%5E%28-4%29
and our equation becomes:
3q%5E2+-21q+%2B+3+=+0
We can start by trying to factor. We can factor out a 3:
3%28q%5E2+-7q+%2B+1%29+=+0
The trinomial will not factor. So we will use the Quadratic formula on q%5E2+-7q+%2B+1:
q+=+%28-%28-7%29+%2B-+sqrt%28%28-7%29%5E2+-+4%281%29%281%29%29%29%2F2%281%29
which simplifies as follows:
q+=+%287+%2B-+sqrt%2849+-+4%281%29%281%29%29%29%2F2
q+=+%287+%2B-+sqrt%2849+-+4%29%29%2F2
q+=+%287+%2B-+sqrt%2845%29%29%2F2
q+=+%287+%2B-+sqrt%289%2A5%29%29%2F2
q+=+%287+%2B-+sqrt%289%29%2Asqrt%285%29%29%2F2
q+=+%287+%2B-+3sqrt%285%29%29%2F2
This is a solution for q. But we're not interested in what q is. We are interested in the solutions for x. This is where we replace q with x%5E%28-2%29:
x%5E%28-2%29+=+%287+%2B-+3sqrt%285%29%29%2F2
Since x%5E%28-2%29+=+1%2Fx%5E2:
1%2Fx%5E2+=+%287+%2B-+3sqrt%285%29%29%2F2
We can flip both sides updies down. (If two numbers are equal then their reciprocals are equal.):
x%5E2+=+2%2F%287+%2B-+3sqrt%285%29%29
Find the square root of each side:
sqrt%28x%5E2%29+=+sqrt%282%2F%287+%2B-+3sqrt%285%29%29%29
abs%28x%29+=+sqrt%282%2F%287+%2B-+3sqrt%285%29%29%29
x+=+0+%2B-+sqrt%282%2F%287+%2B-+3sqrt%285%29%29%29 (Note: Algebra.com's formula software cannot do "+-" without something in front of it. So I added a 0 for this reason. The 0 is really optional.)

This may be an acceptable answer. With the two sets of "+-", we have 4 different solutions for x:
  • x+=+sqrt%282%2F%287+%2B+3sqrt%285%29%29%29
  • x+=+sqrt%282%2F%287+-+3sqrt%285%29%29%29
  • x+=+-sqrt%282%2F%287+%2B+3sqrt%285%29%29%29
  • x+=+-sqrt%282%2F%287+-+3sqrt%285%29%29%29

These solutions, however, have square roots in the denominator and we don't usually leave square roots in denominators. If you feel you must rationalize the denominators I'll give you some clues on how to do so:
  • For the 7%2B3sqrt%285%29 denominators, multiply the numerator and denominator of the fraction by 7-3sqrt%285%29.
  • For the 7-3sqrt%285%29 denominators, multiply the numerator and denominator of the fraction by 7%2B3sqrt%285%29.

Multiplying 7%2B3sqrt%285%29 and 7-3sqrt%285%29 in the denominators will result in: 7%5E2+-+%283sqrt%285%29%29%5E2+=+49+-+45+=+4 which is a nice rational number.

After doing several of this kind of problem, you will learn how to solve them without a temporary variable. You will see how to go from
3%2Ax%5E%28-4%29-21%2Ax%5E-2%2B3+=+0
to
3%28x%5E%28-4%29-7%2Ax%5E%28-2%29%2B1%29+=+0
and then to the Quadratic formula:
x%5E%28-2%29+=+%28-%28-7%29+%2B-+sqrt%28%28-7%29%5E2+-+4%281%29%281%29%29%29%2F2%281%29
etc. without the use of any temporary variable.