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tester [92]
3 years ago
7

Solving Rational Functions Hello I'm posting again because I really need help on this any help is appreciated!!​

Mathematics
1 answer:
Greeley [361]3 years ago
6 0

Answer:

x = √17 and x = -√17

Step-by-step explanation:

We have the equation:

\frac{3}{x + 4}  - \frac{1}{x + 3}  = \frac{x + 9}{(x^2 + 7x + 12)}

To solve this we need to remove the denominators.

Then we can first multiply both sides by (x + 4) to get:

\frac{3*(x + 4)}{x + 4}  - \frac{(x + 4)}{x + 3}  = \frac{(x + 9)*(x + 4)}{(x^2 + 7x + 12)}

3  - \frac{(x + 4)}{x + 3}  = \frac{(x + 9)*(x + 4)}{(x^2 + 7x + 12)}

Now we can multiply both sides by (x + 3)

3*(x + 3)  - \frac{(x + 4)*(x+3)}{x + 3}  = \frac{(x + 9)*(x + 4)*(x+3)}{(x^2 + 7x + 12)}

3*(x + 3)  - (x + 4)  = \frac{(x + 9)*(x + 4)*(x+3)}{(x^2 + 7x + 12)}

(2*x + 5)  = \frac{(x + 9)*(x + 4)*(x+3)}{(x^2 + 7x + 12)}

Now we can multiply both sides by (x^2 + 7*x + 12)

(2*x + 5)*(x^2 + 7x + 12)  = \frac{(x + 9)*(x + 4)*(x+3)}{(x^2 + 7x + 12)}*(x^2 + 7x + 12)

(2*x + 5)*(x^2 + 7x + 12)  = (x + 9)*(x + 4)*(x+3)

Now we need to solve this:

we will get

2*x^3 + 19*x^2 + 59*x + 60 =  (x^2 + 13*x + 3)*(x + 3)

2*x^3 + 19*x^2 + 59*x + 60 =  x^3 + 16*x^2 + 42*x + 9

Then we get:

2*x^3 + 19*x^2 + 59*x + 60 - (  x^3 + 16*x^2 + 42*x + 9) = 0

x^3 + 3x^2 + 17*x + 51 = 0

So now we only need to solve this.

We can see that the constant is 51.

Then one root will be a factor of 51.

The factors of -51 are:

-3 and -17

Let's try -3

p( -3) = (-3)^3 + 3*(-3)^2 + +17*(-3) + 51 = 0

Then x = -3 is one solution of the equation.

But if we look at the original equation, x = -3 will lead to a zero in one denominator, then this solution can be ignored.

This means that we can take a factor (x + 3) out, so we can rewrite our equation as:

x^3 + 3x^2 + 17*x + 51 = (x + 3)*(x^2 + 17) = 0

The other two solutions are when the other term is equal to zero.

Then the other two solutions are given by:

x = ±√17

And neither of these have problems in the denominators, so we can conclude that the solutions are:

x = √17 and x = -√17

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  6  

 —————

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Step-by-step explanation:

Step by Step Solution:

More Icon

STEP

1

:

Equation at the end of step 1

 ((12•(n3))-(24•(n2)))       (12n-42)      

 —————————————————————•———————————————————

  (((4•(n2))-22n)+28)  ((6•(n3))+(24•3n2))  

STEP  

2

:

Equation at the end of step

2

:

 ((12•(n3))-(24•(n2)))      (12n-42)      

 —————————————————————•——————————————————

  (((4•(n2))-22n)+28)  ((2•3n3)+(24•3n2))  

STEP

3

:

            12n - 42  

Simplify   ——————————

           6n3 + 48n2

STEP

4

:

Pulling out like terms

4.1     Pull out like factors :

  12n - 42  =   6 • (2n - 7)  

STEP

5

:

Pulling out like terms

5.1     Pull out like factors :

  6n3 + 48n2  =   6n2 • (n + 8)  

Equation at the end of step

5

:

 ((12•(n3))-(24•(n2)))  (2n-7)  

 —————————————————————•————————

  (((4•(n2))-22n)+28)  n2•(n+8)

STEP  

6

:

Equation at the end of step

6

:

 ((12•(n3))-(24•(n2)))  (2n-7)  

 —————————————————————•————————

    ((22n2-22n)+28)    n2•(n+8)

STEP  

7

:

Equation at the end of step

7

:

 ((12•(n3))-(23•3n2))  (2n-7)  

 ————————————————————•————————

     (4n2-22n+28)     n2•(n+8)

STEP  

8

:

Equation at the end of step

8

:

 ((22•3n3) - (23•3n2))      (2n - 7)  

 ————————————————————— • ————————————

   (4n2 - 22n + 28)      n2 • (n + 8)

STEP

9

:

             12n3 - 24n2  

Simplify   ——————————————

           4n2 - 22n + 28

STEP

10

:

Pulling out like terms

10.1     Pull out like factors :

  12n3 - 24n2  =   12n2 • (n - 2)  

STEP

11

:

Pulling out like terms

11.1     Pull out like factors :

  4n2 - 22n + 28  =   2 • (2n2 - 11n + 14)  

Trying to factor by splitting the middle term

11.2     Factoring  2n2 - 11n + 14  

The first term is,  2n2  its coefficient is  2 .

The middle term is,  -11n  its coefficient is  -11 .

The last term, "the constant", is  +14  

Step-1 : Multiply the coefficient of the first term by the constant   2 • 14 = 28  

Step-2 : Find two factors of  28  whose sum equals the coefficient of the middle term, which is   -11 .

     -28    +    -1    =    -29  

     -14    +    -2    =    -16  

     -7    +    -4    =    -11    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -7  and  -4  

                    2n2 - 7n - 4n - 14

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

                   n • (2n-7)

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

                   2 • (2n-7)

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

                   (n-2)  •  (2n-7)

            Which is the desired factorization

Canceling Out :

11.3    Cancel out  (n-2)  which appears on both sides of the fraction line.

Equation at the end of step

11

:

   6n2      (2n - 7)  

 —————— • ————————————

 2n - 7   n2 • (n + 8)

STEP

12

:

Canceling Out

12.1    Cancel out  (2n-7)  which appears on both sides of the fraction line.

Canceling Out :

12.2    Canceling out n2 as it appears on both sides of the fraction line

Final result :

   6  

 —————

 n + 8

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