How do you solve #12/(a+3)+6/(a^2-9)=8/(a+3)#?

Question

Genesis Allen · Accepted Answer

#color(blue)(x=3/2)#

#12/(a+3)+6/(a^2-9)=8/(a+3)#

factor #(a^2-9)#

This is the difference of two squares:

#(a^2-b^2)=(a+b)(a-b)#

#(a^2-3^2)=(a+3)(a-3)#

#12/(a+3)+6/((a+3)(a-3))=8/(a+3)#

Multiply through by #(x+3)#

#(a+3)12/(a+3)+(a+3)6/((a+3)(a-3))=(a+3)8/(a+3)#

Cancel:

#cancel((a+3))12/cancel((a+3))+cancel((a+3))6/(cancel((a+3))(a-3))=cancel((a+3))8/cancel((a+3))#

#12+6/(x-3)=8#

Subtract #12#

#6/(x-3)=-4#

Multiply by #(x-3)#

#(x-3)6/(x-3)=-4(x-3)#

Cancel:

#cancel((x-3))6/(cancel(x-3))=-4(x-3)#

#6=-4(x-3)#

Divide by #-4#

#x-3=6/-4#

#x=6/-4+3=3/2#

#color(blue)(x=3/2)#

Genesis Allen · Answer

undefined To solve the equation 12/(a+3) + 6/(a^2-9) = 8/(a+3), we can follow these steps:

1. Start by simplifying the equation. The denominator a^2-9 can be factored as (a+3)(a-3), so we can rewrite the equation as:
   12/(a+3) + 6/[(a+3)(a-3)] = 8/(a+3)

2. Next, we can eliminate the denominators by multiplying every term in the equation by (a+3)(a-3). This gives us:
   12(a-3) + 6 = 8(a-3)

3. Distribute and simplify the equation:
   12a - 36 + 6 = 8a - 24

4. Combine like terms:
   12a - 30 = 8a - 24

5. Move all the terms involving 'a' to one side of the equation by subtracting 8a from both sides:
   12a - 8a - 30 = -24

6. Simplify the equation further:
   4a - 30 = -24

7. Move the constant term to the other side by adding 30 to both sides:
   4a - 30 + 30 = -24 + 30

8. Simplify the equation:
   4a = 6

9. Solve for 'a' by dividing both sides by 4:
   4a/4 = 6/4

10. Simplify the equation:
    a = 3/2

Therefore, the solution to the equation 12/(a+3) + 6/(a^2-9) = 8/(a+3) is a = 3/2.