Question

find the roots of the following x²-12x+27=0​

Answer 1

$\huge\rm\underline\purple{Answer:-}$

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 That's it

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

x2 - 9x - 3x - 27

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x-9)

Add up the last 2 terms, pulling out common factors :

3 • (x-9)

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

(x-3) • (x-9)

Which is the desired factorization

Equation at the end of step 1 :

(x - 3) • (x - 9) = 0

STEP 2:

Theory - Roots of a product

2.1 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:

2.2 Solve : x-3 = 0

Add 3 to both sides of the equation :

x = 3

Solving a Single Variable Equation:

2.3 Solve : x-9 = 0

Add 9 to both sides of the equation :

x = 9

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Answer 2

Answer:

9,3

Step-by-step explanation:

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

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

                    x2 - 9x - 3x - 27

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x • (x-9)

             Add up the last 2 terms, pulling out common factors :

                   3 • (x-9)

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

                   (x-3)  •  (x-9)

            Which is the desired factorization

                   (x - 3) • (x - 9)  = 0

                  Therefore factor of x = 9 , 3

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