Question

middle term split the equation 3 X square + 15 x

Answer 1

3x2-15x+18=0 Two solutions were found : x = 3 x = 2

Step by step solution :Step  1  :Equation at the end of step  1  : (3x2 - 15x) + 18 = 0 Step  2  :Step  3  :Pulling out like terms :

 3.1     Pull out like factors :

   3x2 - 15x + 18  =   3 • (x2 - 5x + 6) 

Trying to factor by splitting the middle term

 3.2     Factoring  x2 - 5x + 6 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -5x  its coefficient is  -5 .
The last term, "the constant", is  +6 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 6 = 6 

Step-2 : Find two factors of  6  whose sum equals the coefficient of the middle term, which is   -5 .

     -6   +   -1   =   -7     -3   +   -2   =   -5   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -3  and  -2 
                     x2 - 3x - 2x - 6

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-3)
              Add up the last 2 terms, pulling out common factors :
                    2 • (x-3)
Step-5 : Add up the four terms of step 4 :
                    (x-2)  •  (x-3)
             Which is the desired factorization

Equation at the end of step  3  : 3 • (x - 2) • (x - 3) = 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.

Equations which are never true :

 4.2      Solve :    3   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.3      Solve  :    x-2 = 0 

 Add  2  to both sides of the equation : 
                      x = 2 

Solving a Single Variable Equation :

 4.4      Solve  :    x-3 = 0 

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

Supplement : Solving Quadratic Equation DirectlySolving  x2-5x+6  = 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

Answer 2

$3x {}^{2} + 15x$

$3x(x + 5)$

$x = 0 \: \: and \: \: \: - 5$


Thanks