Math, asked by karrivulavenkaiah31, 10 months ago

the roots of the equation: 6(x²+1/x²)-25(x-1/x)+12=0​

Answers

Answered by kasiram569
0

Answer:

6(x2+1/x2)-25(x-1/x)+12=0  

Four solutions were found :

x = 3

x = 2

x = -1/2 = -0.500

x = -1/3 = -0.333

Step by step solution :

Step  1  :

           1

Simplify   —

           x

Equation at the end of step  1  :

             1           1

 ((6•((x2)+————))-(25•(x-—)))+12  = 0  

           (x2)          x

Step  2  :

Rewriting the whole as an Equivalent Fraction :

2.1   Subtracting a fraction from a whole

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

         x     x • x

    x =  —  =  —————

         1       x  

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:

x • x - (1)     x2 - 1

———————————  =  ——————

     x            x    

Equation at the end of step  2  :

             1        (x2-1)

 ((6•((x2)+————))-(25•——————))+12  = 0  

           (x2)         x    

Step  3  :

Trying to factor as a Difference of Squares :

3.1      Factoring:  x2-1  

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

        A2 - AB + BA - B2 =

        A2 - AB + AB - B2 =

        A2 - B2

Step  4  :

            1

Simplify   ——

           x2

Equation at the end of step  4  :

            1   25•(x+1)•(x-1)

 ((6•((x2)+——))-——————————————)+12  = 0  

           x2         x        

          x2     x2 • x2

    x2 =  ——  =  ———————

          1        x2    

x2 • x2 + 1     x4 + 1

———————————  =  ——————

    x2            x2  

Equation at the end of step  5  :

     (x4+1)  25•(x+1)•(x-1)

 ((6•——————)-——————————————)+12  = 0  

       x2          x        

Step  6  :

Polynomial Roots Calculator :

6.1    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 ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        2.00      

     1       1        1.00        2.00      

Polynomial Roots Calculator found no rational roots

Equation at the end of step  6  :

  6•(x4+1) 25•(x+1)•(x-1)

 (————————-——————————————)+12  = 0  

     x2          x        

Step  7  :

Calculating the Least Common Multiple :

7.1    Find the Least Common Multiple

     The left denominator is :       x2  

     The right denominator is :       x  

                 Number of times each Algebraic Factor

           appears in the factorization of:

   Algebraic    

   Factor      Left  

Denominator   Right  

Denominator   L.C.M = Max  

{Left,Right}  

x  2 1 2

     Least Common Multiple:

     x2  

Calculating Multipliers :

7.2    Calculate multipliers for the two fractions

   Denote the Least Common Multiple by  L.C.M  

   Denote the Left Multiplier by  Left_M  

   Denote the Right Multiplier by  Right_M  

   Denote the Left Deniminator by  L_Deno  

   Denote the Right Multiplier by  R_Deno  

  Left_M = L.C.M / L_Deno = 1

  Right_M = L.C.M / R_Deno = x

Equation at the end of step  8  :

 (3x + 1) • (2x + 1) • (x - 2) • (x - 3)

 ———————————————————————————————————————  = 0  

                   x2                    

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