3x2+36x+81 = 0
Solving quadratic formula

STEP

1

:

Equation at the end of step 1

 (3x2 +  36x) +  81  = 0

STEP

2

:

STEP

3

:

Pulling out like terms

3.1     Pull out like factors :

  3x2 + 36x + 81  =   3 • (x2 + 12x + 27)

Trying to factor by splitting the middle term

3.2     Factoring  x2 + 12x + 27

The first term is,  x2  its coefficient is  1 .

The middle term is,  +12x  its coefficient is  12 .

The last term, "the constant", is  +27

Step-1 : Multiply the coefficient of the first term by the constant   1 • 27 = 27

Step-2 : Find two factors of  27  whose sum equals the coefficient of the middle term, which is   12 .

     -27    +    -1    =    -28

     -9    +    -3    =    -12

     -3    +    -9    =    -12

     -1    +    -27    =    -28

     1    +    27    =    28

     3    +    9    =    12    That's it

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

                    x2 + 3x + 9x + 27

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 :

                   9 • (x+3)

Step-5 : Add up the four terms of step 4 :

                   (x+9)  •  (x+3)

            Which is the desired factorization

Equation at the end of step

3

:

 3 • (x + 9) • (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+9 = 0

Subtract  9  from both sides of the equation :

                     x = -9

Solving a Single Variable Equation:

4.4      Solve  :    x+3 = 0

Subtract  3  from both sides of the equation :

                     x = -3

Supplement : Solving Quadratic Equation Directly

Solving    x2+12x+27  = 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

Step-by-step explanation:

the answer is here: : )

:

good luck on your studies! : )

david edward.