Solution - Factoring binomials using the difference of squares

Rearrange:

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

                     -7*u^3*u-(-5-19)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (0 -  (7u3 • u)) -  -24  = 0 

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  24-7u⁴ 

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 :  24  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(u) = -7u⁴+24
Polynomial Roots Calculator is a set of methods aimed at finding values of  u  for which   F(u)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  u  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  24  and the Trailing Constant is  -7.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,4 ,6 ,8 ,12 ,24
 of the Trailing Constant :  1 ,7

 Let us test ....

Note - For tidiness, printing of 27 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

  24 - 7u4  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    -7u⁴+24 = 0 

 Subtract  24  from both sides of the equation : 
                      -7u⁴ = -24
Multiply both sides of the equation by (-1) :  7u⁴ = 24


Divide both sides of the equation by 7:
                     u⁴ = 24/7 = 3.429
                     u  =  ∜ 24/7  

 The equation has two real solutions  
 These solutions are  u = ∜ 3.429 = ± 1.36075  
 

Two solutions were found :