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[(x^2 -3x +2)/(x^2 -5x +4)] - [(x^2 + 10x + 24)/(x^2 + 8x + 12)]

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Sagot :

Answer:

x-8/x-6

Step-by-step explanation:

Step by Step Solution

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STEP

1

:

              8x - 32  

Simplify   —————————————

           x2 - 10x + 24

STEP

2

:

Pulling out like terms

2.1     Pull out like factors :

  8x - 32  =   8 • (x - 4)

Trying to factor by splitting the middle term

2.2     Factoring  x2 - 10x + 24

The first term is,  x2  its coefficient is  1 .

The middle term is,  -10x  its coefficient is  -10 .

The last term, "the constant", is  +24

Step-1 : Multiply the coefficient of the first term by the constant   1 • 24 = 24

Step-2 : Find two factors of  24  whose sum equals the coefficient of the middle term, which is   -10 .

     -24    +    -1    =    -25

     -12    +    -2    =    -14

     -8    +    -3    =    -11

     -6    +    -4    =    -10    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  -4

                    x2 - 6x - 4x - 24

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

                   x • (x-6)

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

                   4 • (x-6)

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

                   (x-4)  •  (x-6)

            Which is the desired factorization

Canceling Out :

2.3    Cancel out  (x-4)  which appears on both sides of the fraction line.

Equation at the end of step

2

:

    ((x2)-4x)     8

 ———————————————-———

 (((x2)-10x)+24) x-6

STEP

3

:

              x2 - 4x  

Simplify   —————————————

           x2 - 10x + 24

STEP

4

:

Pulling out like terms

4.1     Pull out like factors :

  x2 - 4x  =   x • (x - 4)

Trying to factor by splitting the middle term

4.2     Factoring  x2 - 10x + 24

The first term is,  x2  its coefficient is  1 .

The middle term is,  -10x  its coefficient is  -10 .

The last term, "the constant", is  +24

Step-1 : Multiply the coefficient of the first term by the constant   1 • 24 = 24

Step-2 : Find two factors of  24  whose sum equals the coefficient of the middle term, which is   -10 .

     -24    +    -1    =    -25

     -12    +    -2    =    -14

     -8    +    -3    =    -11

     -6    +    -4    =    -10    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  -4

                    x2 - 6x - 4x - 24

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

                   x • (x-6)

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

                   4 • (x-6)

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

                   (x-4)  •  (x-6)

            Which is the desired factorization

Canceling Out :

4.3    Cancel out  (x-4)  which appears on both sides of the fraction line.

Equation at the end of step

4

:

   x        8  

 ————— -  —————

 x - 6    x - 6

STEP

5

:

Adding fractions which have a common denominator :

5.1       Adding fractions which have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x - (8)     x - 8

———————  =  —————

  x-6       x - 6

Final result :

 x - 8

 —————

 x - 6

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