# Solve for n 4/(n-3)=3/n+2 4n-3=3n+2
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.
n-3,n,1
Since n-3,n,1 contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for n-3,n,1 are:
1. Find the LCM for the numeric part 1,1,1.
2. Find the LCM for the variable part n1.
3. Find the LCM for the compound variable part n-3.
4. Multiply each LCM together.
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 n1 is n itself.
n1=n
n occurs 1 time.
The LCM of n1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
n
The factor for n-3 is n-3 itself.
(n-3)=n-3
(n-3) occurs 1 time.
The LCM of n-3 is the result of multiplying all factors the greatest number of times they occur in either term.
n-3
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
n(n-3)
n(n-3)
Multiply each term by n(n-3) and simplify.
Multiply each term in 4n-3=3n+2 by n(n-3) in order to remove all the denominators from the equation.
4n-3⋅(n(n-3))=3n⋅(n(n-3))+2⋅(n(n-3))
Cancel the common factor of n-3.
Factor n-3 out of n(n-3).
4n-3⋅((n-3)n)=3n⋅(n(n-3))+2⋅(n(n-3))
Cancel the common factor.
4n-3⋅((n-3)n)=3n⋅(n(n-3))+2⋅(n(n-3))
Rewrite the expression.
4⋅n=3n⋅(n(n-3))+2⋅(n(n-3))
4n=3n⋅(n(n-3))+2⋅(n(n-3))
Simplify 3n⋅(n(n-3))+2⋅(n(n-3)).
Simplify each term.
Cancel the common factor of n.
Cancel the common factor.
4n=3n⋅(n(n-3))+2⋅(n(n-3))
Rewrite the expression.
4n=3⋅(n-3)+2⋅(n(n-3))
4n=3⋅(n-3)+2⋅(n(n-3))
Apply the distributive property.
4n=3n+3⋅-3+2⋅(n(n-3))
Multiply 3 by -3.
4n=3n-9+2⋅(n(n-3))
Apply the distributive property.
4n=3n-9+2⋅(n⋅n+n⋅-3)
Multiply n by n.
4n=3n-9+2⋅(n2+n⋅-3)
Move -3 to the left of n.
4n=3n-9+2⋅(n2-3⋅n)
Apply the distributive property.
4n=3n-9+2n2+2(-3n)
Multiply -3 by 2.
4n=3n-9+2n2-6n
4n=3n-9+2n2-6n
Subtract 6n from 3n.
4n=-3n-9+2n2
4n=-3n-9+2n2
4n=-3n-9+2n2
Solve the equation.
Since n is on the right side of the equation, switch the sides so it is on the left side of the equation.
-3n-9+2n2=4n
Move all terms containing n to the left side of the equation.
Subtract 4n from both sides of the equation.
-3n-9+2n2-4n=0
Subtract 4n from -3n.
-7n-9+2n2=0
-7n-9+2n2=0
Factor by grouping.
Reorder terms.
2n2-7n-9=0
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=2⋅-9=-18 and whose sum is b=-7.
Factor -7 out of -7n.
2n2-7n-9=0
Rewrite -7 as 2 plus -9
2n2+(2-9)n-9=0
Apply the distributive property.
2n2+2n-9n-9=0
2n2+2n-9n-9=0
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(2n2+2n)-9n-9=0
Factor out the greatest common factor (GCF) from each group.
2n(n+1)-9(n+1)=0
2n(n+1)-9(n+1)=0
Factor the polynomial by factoring out the greatest common factor, n+1.
(n+1)(2n-9)=0
(n+1)(2n-9)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
n+1=0
2n-9=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
n+1=0
Subtract 1 from both sides of the equation.
n=-1
n=-1
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
2n-9=0
Add 9 to both sides of the equation.
2n=9
Divide each term by 2 and simplify.
Divide each term in 2n=9 by 2.
2n2=92
Cancel the common factor of 2.
Cancel the common factor.
2n2=92
Divide n by 1.
n=92
n=92
n=92
n=92
The final solution is all the values that make (n+1)(2n-9)=0 true.
n=-1,92
n=-1,92
The result can be shown in multiple forms.
Exact Form:
n=-1,92
Decimal Form:
n=-1,4.5
Mixed Number Form:
n=-1,412
Solve for n 4/(n-3)=3/n+2

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