# Solve for a 2/(5-a)=(a+4)/(a-5)

25-a=a+4a-5
Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.
2⋅(a-5)=(5-a)⋅(a+4)
Solve the equation for a.
Since a is on the right side of the equation, switch the sides so it is on the left side of the equation.
(5-a)⋅(a+4)=2⋅(a-5)
Simplify (5-a)⋅(a+4).
Expand (5-a)(a+4) using the FOIL Method.
Apply the distributive property.
5(a+4)-a(a+4)=2⋅(a-5)
Apply the distributive property.
5a+5⋅4-a(a+4)=2⋅(a-5)
Apply the distributive property.
5a+5⋅4-a⋅a-a⋅4=2⋅(a-5)
5a+5⋅4-a⋅a-a⋅4=2⋅(a-5)
Simplify and combine like terms.
Simplify each term.
Multiply 5 by 4.
5a+20-a⋅a-a⋅4=2⋅(a-5)
Multiply a by a by adding the exponents.
Move a.
5a+20-(a⋅a)-a⋅4=2⋅(a-5)
Multiply a by a.
5a+20-a2-a⋅4=2⋅(a-5)
5a+20-a2-a⋅4=2⋅(a-5)
Multiply 4 by -1.
5a+20-a2-4a=2⋅(a-5)
5a+20-a2-4a=2⋅(a-5)
Subtract 4a from 5a.
a+20-a2=2⋅(a-5)
a+20-a2=2⋅(a-5)
a+20-a2=2⋅(a-5)
Simplify 2⋅(a-5).
Apply the distributive property.
a+20-a2=2a+2⋅-5
Multiply 2 by -5.
a+20-a2=2a-10
a+20-a2=2a-10
Move all terms containing a to the left side of the equation.
Subtract 2a from both sides of the equation.
a+20-a2-2a=-10
Subtract 2a from a.
-a+20-a2=-10
-a+20-a2=-10
Move 10 to the left side of the equation by adding it to both sides.
-a+20-a2+10=0
-a-a2+30=0
Factor the left side of the equation.
Factor -1 out of -a-a2+30.
Reorder -a and -a2.
-a2-a+30=0
Factor -1 out of -a2.
-(a2)-a+30=0
Factor -1 out of -a.
-(a2)-(a)+30=0
Rewrite 30 as -1(-30).
-(a2)-(a)-1⋅-30=0
Factor -1 out of -(a2)-(a).
-(a2+a)-1⋅-30=0
Factor -1 out of -(a2+a)-1(-30).
-(a2+a-30)=0
-(a2+a-30)=0
Factor.
Factor a2+a-30 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -30 and whose sum is 1.
-5,6
Write the factored form using these integers.
-((a-5)(a+6))=0
-((a-5)(a+6))=0
Remove unnecessary parentheses.
-(a-5)(a+6)=0
-(a-5)(a+6)=0
-(a-5)(a+6)=0
Multiply each term in -(a-5)(a+6)=0 by -1
Multiply each term in -(a-5)(a+6)=0 by -1.
(-(a-5)(a+6))⋅-1=0⋅-1
Simplify (-(a-5)(a+6))⋅-1.
Simplify by multiplying through.
Apply the distributive property.
(-a–5)(a+6)⋅-1=0⋅-1
Multiply -1 by -5.
(-a+5)(a+6)⋅-1=0⋅-1
(-a+5)(a+6)⋅-1=0⋅-1
Expand (-a+5)(a+6) using the FOIL Method.
Apply the distributive property.
(-a(a+6)+5(a+6))⋅-1=0⋅-1
Apply the distributive property.
(-a⋅a-a⋅6+5(a+6))⋅-1=0⋅-1
Apply the distributive property.
(-a⋅a-a⋅6+5a+5⋅6)⋅-1=0⋅-1
(-a⋅a-a⋅6+5a+5⋅6)⋅-1=0⋅-1
Simplify and combine like terms.
Simplify each term.
Multiply a by a by adding the exponents.
Move a.
(-(a⋅a)-a⋅6+5a+5⋅6)⋅-1=0⋅-1
Multiply a by a.
(-a2-a⋅6+5a+5⋅6)⋅-1=0⋅-1
(-a2-a⋅6+5a+5⋅6)⋅-1=0⋅-1
Multiply 6 by -1.
(-a2-6a+5a+5⋅6)⋅-1=0⋅-1
Multiply 5 by 6.
(-a2-6a+5a+30)⋅-1=0⋅-1
(-a2-6a+5a+30)⋅-1=0⋅-1
(-a2-a+30)⋅-1=0⋅-1
(-a2-a+30)⋅-1=0⋅-1
Apply the distributive property.
-a2⋅-1-a⋅-1+30⋅-1=0⋅-1
Simplify.
Multiply -a2⋅-1.
Multiply -1 by -1.
1a2-a⋅-1+30⋅-1=0⋅-1
Multiply a2 by 1.
a2-a⋅-1+30⋅-1=0⋅-1
a2-a⋅-1+30⋅-1=0⋅-1
Multiply -a⋅-1.
Multiply -1 by -1.
a2+1a+30⋅-1=0⋅-1
Multiply a by 1.
a2+a+30⋅-1=0⋅-1
a2+a+30⋅-1=0⋅-1
Multiply 30 by -1.
a2+a-30=0⋅-1
a2+a-30=0⋅-1
a2+a-30=0⋅-1
Multiply 0 by -1.
a2+a-30=0
a2+a-30=0
Factor a2+a-30 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -30 and whose sum is 1.
-5,6
Write the factored form using these integers.
(a-5)(a+6)=0
(a-5)(a+6)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
a-5=0
a+6=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
a-5=0
Add 5 to both sides of the equation.
a=5
a=5
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
a+6=0
Subtract 6 from both sides of the equation.
a=-6
a=-6
The final solution is all the values that make (a-5)(a+6)=0 true.
a=5,-6
a=5,-6
Exclude the solutions that do not make 25-a=a+4a-5 true.
a=-6
Solve for a 2/(5-a)=(a+4)/(a-5)

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