# Solve for a 2/(a-3)+5/(a+4)=1

2a-3+5a+4=1
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
a-3,a+4,1
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for a-3 is a-3 itself.
(a-3)=a-3
(a-3) occurs 1 time.
The factor for a+4 is a+4 itself.
(a+4)=a+4
(a+4) occurs 1 time.
The LCM of a-3,a+4 is the result of multiplying all factors the greatest number of times they occur in either term.
(a-3)(a+4)
(a-3)(a+4)
Multiply each term by (a-3)(a+4) and simplify.
Multiply each term in 2a-3+5a+4=1 by (a-3)(a+4) in order to remove all the denominators from the equation.
2a-3⋅((a-3)(a+4))+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
Simplify 2a-3⋅((a-3)(a+4))+5a+4⋅((a-3)(a+4)).
Simplify each term.
Cancel the common factor of a-3.
Cancel the common factor.
2a-3⋅((a-3)(a+4))+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
Rewrite the expression.
2⋅(a+4)+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
2⋅(a+4)+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
Apply the distributive property.
2a+2⋅4+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
Multiply 2 by 4.
2a+8+5a+4⋅((a-3)(a+4))=1⋅((a-3)(a+4))
Cancel the common factor of a+4.
Factor a+4 out of (a-3)(a+4).
2a+8+5a+4⋅((a+4)(a-3))=1⋅((a-3)(a+4))
Cancel the common factor.
2a+8+5a+4⋅((a+4)(a-3))=1⋅((a-3)(a+4))
Rewrite the expression.
2a+8+5⋅(a-3)=1⋅((a-3)(a+4))
2a+8+5⋅(a-3)=1⋅((a-3)(a+4))
Apply the distributive property.
2a+8+5a+5⋅-3=1⋅((a-3)(a+4))
Multiply 5 by -3.
2a+8+5a-15=1⋅((a-3)(a+4))
2a+8+5a-15=1⋅((a-3)(a+4))
7a+8-15=1⋅((a-3)(a+4))
Subtract 15 from 8.
7a-7=1⋅((a-3)(a+4))
7a-7=1⋅((a-3)(a+4))
7a-7=1⋅((a-3)(a+4))
Simplify 1⋅((a-3)(a+4)).
Multiply (a-3)(a+4) by 1.
7a-7=(a-3)(a+4)
Expand (a-3)(a+4) using the FOIL Method.
Apply the distributive property.
7a-7=a(a+4)-3(a+4)
Apply the distributive property.
7a-7=a⋅a+a⋅4-3(a+4)
Apply the distributive property.
7a-7=a⋅a+a⋅4-3a-3⋅4
7a-7=a⋅a+a⋅4-3a-3⋅4
Simplify and combine like terms.
Simplify each term.
Multiply a by a.
7a-7=a2+a⋅4-3a-3⋅4
Move 4 to the left of a.
7a-7=a2+4⋅a-3a-3⋅4
Multiply -3 by 4.
7a-7=a2+4a-3a-12
7a-7=a2+4a-3a-12
Subtract 3a from 4a.
7a-7=a2+a-12
7a-7=a2+a-12
7a-7=a2+a-12
7a-7=a2+a-12
Solve the equation.
Since a is on the right side of the equation, switch the sides so it is on the left side of the equation.
a2+a-12=7a-7
Move all terms containing a to the left side of the equation.
Subtract 7a from both sides of the equation.
a2+a-12-7a=-7
Subtract 7a from a.
a2-6a-12=-7
a2-6a-12=-7
Move all terms to the left side of the equation and simplify.
Move 7 to the left side of the equation by adding it to both sides.
a2-6a-12+7=0
a2-6a-5=0
a2-6a-5=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=1, b=-6, and c=-5 into the quadratic formula and solve for a.
6±(-6)2-4⋅(1⋅-5)2⋅1
Simplify.
Simplify the numerator.
Raise -6 to the power of 2.
a=6±36-4⋅(1⋅-5)2⋅1
Multiply -5 by 1.
a=6±36-4⋅-52⋅1
Multiply -4 by -5.
a=6±36+202⋅1
a=6±562⋅1
Rewrite 56 as 22⋅14.
Factor 4 out of 56.
a=6±4(14)2⋅1
Rewrite 4 as 22.
a=6±22⋅142⋅1
a=6±22⋅142⋅1
Pull terms out from under the radical.
a=6±2142⋅1
a=6±2142⋅1
Multiply 2 by 1.
a=6±2142
Simplify 6±2142.
a=3±14
a=3±14
The final answer is the combination of both solutions.
a=3+14,3-14
a=3+14,3-14
The result can be shown in multiple forms.
Exact Form:
a=3+14,3-14
Decimal Form:
a=6.74165738…,-0.74165738…
Solve for a 2/(a-3)+5/(a+4)=1

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