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+3/x)^2-(16)=0 

Step by step solution :

Step  1  :

            3
 Simplify   —
            x

Equation at the end of step  1  :

         3        
  ((x +  —)2) -  16  = 0 
         x        

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to 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 + 3     x2 + 3
 —————————  =  ——————
     x           x   

Equation at the end of step  2  :

   (x2 + 3)        
  (————————)2) -  16  = 0 
      x            
 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 :

          16     16 • x2
    16 =  ——  =  ———————
          1        x2   

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = (x² + 3)²
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  3.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3

 Let us test ....

Polynomial Roots Calculator found no rational roots

Calculating Multipliers :

 3.3    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 = x²

Adding fractions that have a common denominator :

 3.4       Adding up the two equivalent fractions

 (x2+3)2 - (16 • x2)     x4 - 10x2 + 9
 ———————————————————  =  —————————————
         x2                   x2      

Trying to factor by splitting the middle term

 3.5     Factoring  x⁴ - 10x² + 9 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  -10x²  its coefficient is  -10 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   -10 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -9  and  -1 
                     x⁴ - 9x² - 1x² - 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x² • (x²-9)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x²-9)
Step-5 : Add up the four terms of step 4 :
                    (x²-1)  •  (x²-9)
             Which is the desired factorization

Trying to factor as a Difference of Squares :

 3.6      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

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 1)  •  (x - 1) 

Trying to factor as a Difference of Squares :

 3.7      Factoring:  x² - 9 

Check : 9 is the square of 3
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 3)  •  (x - 3) 

Equation at the end of step  3  :

  (x + 1) • (x - 1) • (x + 3) • (x - 3)
  —————————————————————————————————————  = 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:

  (x+1)•(x-1)•(x+3)•(x-3)
  ——————————————————————— • 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 :
   (x+1)  •  (x-1)  •  (x+3)  •  (x-3)  = 0

Theory - Roots of a product :

 4.2    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 4.3      Solve  :    x+1 = 0 

 Subtract  1  from both sides of the equation : 
                      x = -1

Solving a Single Variable Equation :

 4.4      Solve  :    x-1 = 0 

 Add  1  to both sides of the equation : 
                      x = 1

Solving a Single Variable Equation :

 4.5      Solve  :    x+3 = 0 

 Subtract  3  from both sides of the equation : 
                      x = -3

Solving a Single Variable Equation :

 4.6      Solve  :    x-3 = 0 

 Add  3  to both sides of the equation : 
                      x = 3

Supplement : Solving Quadratic Equation Directly

Solving    x4-10x2+9  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 5.1     Solve   x⁴-10x²+9 = 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 :
 w²-10w+9 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   9.0000  or   1.0000

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
   x⁴-10x²+9 = 0
  are either : 
  x =√ 9.000 = 3.00000  or :
  x =√ 9.000 = -3.00000  or :
  x =√ 1.000 = 1.00000  or :
  x =√ 1.000 = -1.00000

Four solutions were found :

1.  x = 3
2.  x = -3
3.  x = 1
4.  x = -1