How do you solve #(2a-5)/(a-9)+a/(a+9)=-6/(a^2-81)#?

2 Answers
Jul 8, 2017

Solution : #a=3 or a =-13/3#

Explanation:

# (2a-5)/(a-9) +a /(a+9) = - 6/((a^2-81)# or

# (2a-5)/(a-9) +a /(a+9) = - 6/((a+9(a-9)# Multiplying by #(a+9(a-9)# on boh sides we get,

# (2a-5)(a+9) +a(a-9) = - 6# or

# 2a^2 +13a -45 +a^2 -9a +6 =0# or

#3a^2+4a-39=0 or 3a^2+13a-9a-39=0# or

#a(3a+13) -3(3a+13) =0 # or

#(3a+13)(a-3)=0 #, either #3a+13=0 :. a = -13/3# or

#a-3=0:.a =3#

Solution: #a=3 or a =-13/3# [Ans]

Jul 8, 2017

No real solutions and #x=-13/3,x=3# extraneous solutions

Explanation:

#(2a-5)/(a-9)+(a)/(a+9)=- 6/(a^2-81)#

#(2a-5)/(a-9)+(a)/(a+9)+ 6/((a-9)(a+9))=0#

#((a+9)(2a-5)+a(a-9)+6(a+9))/((a-9)(a+9))=0#

Multiply both sides by #(a-9)(a+9)#

#2a^2+13a-45+a^2-9a+6a+54=0#

#3a^2+10a+9=0#

#a=(-b+-sqrt(b^2-4ac))/(2a)#

#a=(-(10)+-sqrt((10)^2-4(3)(9)))/(2(3))#

#a=(-10+-sqrt(100-108))/(2(3))#

#a=(-10+-sqrt(-8))/6#

No real solution because of #sqrt(-8)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Alternate method:

#(2a-5)/(a-9)+(a)/(a+9)=- 6/(a^2-81)#

#(2a-5)/(a-9)+(a)/(a+9)=- 6/((a-9)(a+9))#

#((a+9)(2a-5)+(a-9)a=-6)/((a-9)(a+9))#

Multiply both sides by #(a-9)(a+9)#

#(a+9)(2a-5)+(a-9)a=-6#

#2a^2+13a-45+a^2-9a=-6#

#3a^2+4a-39=0#

#(3a+13)(a-3)=0#

#3a=-13 or a=3#

#a=-13/3 or a=3#

extraneous solutions