Solution - Equations reducible to quadratic form

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((4 • (x4)) -  (5•7x2)) -  9  = 0 
 Step  2  :

Equation at the end of step  2  :

  (22x4 -  (5•7x2)) -  9  = 0 

Step  3  :

Trying to factor by splitting the middle term

 3.1     Factoring  4x⁴-35x²-9 

The first term is,  4x⁴  its coefficient is  4 .
The middle term is,  -35x²  its coefficient is  -35 .
The last term, "the constant", is  -9 

Step-1 : Multiply the coefficient of the first term by the constant   4 • -9 = -36 

Step-2 : Find two factors of  -36  whose sum equals the coefficient of the middle term, which is   -35 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -36  and  1 
                     4x⁴ - 36x² + 1x² - 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    4x² • (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 :
                    (4x²+1)  •  (x²-9)
             Which is the desired factorization

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x²-9 

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

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

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = 4x² + 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  4  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  (x + 3) • (x - 3) • (4x2 + 1)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    x+3 = 0 

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

Solving a Single Variable Equation :

 4.3      Solve  :    x-3 = 0 

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

Solving a Single Variable Equation :

 4.4      Solve  :    4x²+1 = 0 

 Subtract  1  from both sides of the equation : 
                      4x² = -1
Divide both sides of the equation by 4:
                     x² = -1/4 = -0.250
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -1/4  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -1/4  =
                    √ -1• 1/4   =
                    √ -1 •√  1/4   =
                    i •  √ 1/4

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x=  0.0000 + 0.5000 i 
                      x=  0.0000 - 0.5000 i 

Supplement : Solving Quadratic Equation Directly

Solving    4x4-35x2-9  = 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   4x⁴-35x²-9 = 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 :
 4w²-35w-9 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   9.0000  or  -0.2500

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
   4x⁴-35x²-9 = 0
  are either : 
  x =√ 9.000 = 3.00000  or :
  x =√ 9.000 = -3.00000  or :
  x =√-0.250 = 0.0 + 0.50000 i  or :
  x =√-0.250 = 0.0 - 0.50000 i

Four solutions were found :

1.   x=  0.0000 - 0.5000 i 
   
2.   x=  0.0000 + 0.5000 i 
   
3.  x = 3
4.  x = -3