Help in simplifing expressions and solving equations

Question

Simplify the expressions and solve the equations:

 

 

1.  v^2 + v - 12/v^2 + 6v + 8 ÷  2v - 6/v + 2             v^2 + v - 12       ÷      2v -  6

                                                                                 v^2 + 6v +8                 v + 2

 

 

 

 

2.     2__   =   ___ b                   +             b_____

       b -2         b^2 - 3b +  2                     2b - 2

 

 

thank you

Answer 1

2.

2/(b - 2) = b/(b^2 - 3b + 2) + b/(2b - 2)

2/(b - 2) = b/((b - 1)(b - 2)) + b/(2(b - 1))

Note that to avoid division by zero, b ≠ 1 or 2.

Multiply both sides by 2(b - 1)(b - 2):

4(b - 1) = 2b + b(b-2) = b^2

0 = b^2 - 4b + 4 = (b - 2)^2

=> b = 2; BUT IN ORIGINAL EQUATION b CAN’T BE 2,

so there’s no solution.

Answer 2

1.  v²+v-12= v²-3v+4v-12

                 =v(v-3)+4(v-3)

                 =(v+4)(v-3)

 

     v²+6v+8=v²+2v+4v+8

                  =v(v+2)+4(v+2)

                  =(v+4)(v+2)

 

     2v-6=2(v-3)

 

 

    now substitute the above values in the given question and also the divide sign will change into multiply by     taking the reciprocal of the second term

 

 v²+v-12      2v-6       (v+4)(v-3)       (v+2)

————— ÷ ———  =  ——————  · ————                                                                  

v²+6v+8       v+2       (v+4)(v+2)      2(v-3)

 

                              =  1/2   (all the other terms will cancel out)

 

 

 

2.   b²-3b+2=b²-b-2b+2

                  =b(b-1)-2(b-1)

                  =(b-2)(b-1)

 

 

   2           b               b

——— = ————— + ———

 b-2      (b-2)(b-1)    2(b-1)

 

   2          2b+b(b-2)

———= ———————— (taking the lcm of right hand side)

 b-2      2(b-2)(b-1)

 

    2   =  2b+b²-2b

———    ——————

 b-2       2(b-2)(b-1)

 

    2                b²

———    =    ————

     1             2(b-1)

 

4b - 4=b²

 

b²-4b+4=0

(b-2)(b-2)=0

 

b-2=0    b-2=0

b=2, b=2