Solution - Equations reducible to quadratic form

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x^2+1/x^2-(62)=0 

Step by step solution :

Step  1  :

             1
 Simplify   ——
            x2

Equation at the end of step  1  :

            1     
  ((x2) +  ——) -  62  = 0 
           x2     
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  x²  as the denominator :

           x2     x2 • x2
     x2 =  ——  =  ———————
           1        x2   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x2 • x2 + 1     x4 + 1
 ———————————  =  ——————
     x2            x2  

Equation at the end of step  2  :

  (x4 + 1)    
  ———————— -  62  = 0 
     x2       

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          62     62 • x2
    62 =  ——  =  ———————
          1        x2   

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = x⁴ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (x4+1) - (62 • x2)     x4 - 62x2 + 1
 ——————————————————  =  —————————————
         x2                  x2      

Trying to factor by splitting the middle term

 3.4     Factoring  x⁴ - 62x² + 1 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  -62x²  its coefficient is  -62 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1 

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -62 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  3  :

  x4 - 62x2 + 1
  —————————————  = 0 
       x2      

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  x4-62x2+1
  ————————— • x2 = 0 • x2
     x2    

Now, on the left hand side, the  x²  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   x⁴-62x²+1  = 0

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 4.2     Solve   x⁴-62x²+1 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x²  transforms the equation into :
 w²-62w+1 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   61.9839  or   0.0161

Now that we know the value(s) of  w , we can calculate  x  since  x  is  √ w  

Doing just this we discover that the solutions of
   x⁴-62x²+1 = 0
  are either : 
   x =√61.984 = 7.87298  or :
   x =√61.984 = -7.87298  or :
   x =√ 0.016 = 0.12702  or :
   x =√ 0.016 = -0.12702

Four solutions were found :

1.  x =√ 0.016 = -0.12702
2.  x =√ 0.016 = 0.12702
3.  x =√61.984 = -7.87298
4.  x =√61.984 = 7.87298