# Solve for a 1/3*(4a+1)=1/(2a) 13⋅(4a+1)=12a
Simplify 13⋅(4a+1).
Apply the distributive property.
13(4a)+13⋅1=12a
Multiply 13(4a).
Combine 4 and 13.
43a+13⋅1=12a
Combine 43 and a.
4a3+13⋅1=12a
4a3+13⋅1=12a
Multiply 13 by 1.
4a3+13=12a
4a3+13=12a
Subtract 13 from both sides of the equation.
4a3=12a-13
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.
3,2a,3
Since 3,2a,3 contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 3,2,3 then find LCM for the variable part a1.
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.
Since 3 has no factors besides 1 and 3.
3 is a prime number
Since 2 has no factors besides 1 and 2.
2 is a prime number
Since 3 has no factors besides 1 and 3.
3 is a prime number
The LCM of 3,2,3 is the result of multiplying all prime factors the greatest number of times they occur in either number.
2⋅3
Multiply 2 by 3.
6
The factor for a1 is a itself.
a1=a
a occurs 1 time.
The LCM of a1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
a
The LCM for 3,2a,3 is the numeric part 6 multiplied by the variable part.
6a
6a
Multiply each term by 6a and simplify.
Multiply each term in 4a3=12a-13 by 6a in order to remove all the denominators from the equation.
4a3⋅(6a)=12a⋅(6a)-13⋅(6a)
Simplify 4a3⋅(6a).
Rewrite using the commutative property of multiplication.
64a3a=12a⋅(6a)-13⋅(6a)
Cancel the common factor of 3.
Factor 3 out of 6.
3(2)4a3a=12a⋅(6a)-13⋅(6a)
Cancel the common factor.
3⋅24a3a=12a⋅(6a)-13⋅(6a)
Rewrite the expression.
2(4a)a=12a⋅(6a)-13⋅(6a)
2(4a)a=12a⋅(6a)-13⋅(6a)
Multiply 4 by 2.
8a⋅a=12a⋅(6a)-13⋅(6a)
Multiply a by a by adding the exponents.
Move a.
8(a⋅a)=12a⋅(6a)-13⋅(6a)
Multiply a by a.
8a2=12a⋅(6a)-13⋅(6a)
8a2=12a⋅(6a)-13⋅(6a)
8a2=12a⋅(6a)-13⋅(6a)
Simplify each term.
Rewrite using the commutative property of multiplication.
8a2=612aa-13⋅(6a)
Cancel the common factor of 2.
Factor 2 out of 6.
8a2=2(3)12aa-13⋅(6a)
Factor 2 out of 2a.
8a2=2(3)12(a)a-13⋅(6a)
Cancel the common factor.
8a2=2⋅312aa-13⋅(6a)
Rewrite the expression.
8a2=31aa-13⋅(6a)
8a2=31aa-13⋅(6a)
Combine 3 and 1a.
8a2=3aa-13⋅(6a)
Cancel the common factor of a.
Cancel the common factor.
8a2=3aa-13⋅(6a)
Rewrite the expression.
8a2=3-13⋅(6a)
8a2=3-13⋅(6a)
Cancel the common factor of 3.
Move the leading negative in -13 into the numerator.
8a2=3+-13⋅(6a)
Factor 3 out of 6a.
8a2=3+-13⋅(3(2a))
Cancel the common factor.
8a2=3+-13⋅(3(2a))
Rewrite the expression.
8a2=3-1⋅(2a)
8a2=3-1⋅(2a)
Multiply 2 by -1.
8a2=3-2a
8a2=3-2a
8a2=3-2a
Solve the equation.
Add 2a to both sides of the equation.
8a2+2a=3
Move 3 to the left side of the equation by subtracting it from both sides.
8a2+2a-3=0
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=8⋅-3=-24 and whose sum is b=2.
Factor 2 out of 2a.
8a2+2(a)-3=0
Rewrite 2 as -4 plus 6
8a2+(-4+6)a-3=0
Apply the distributive property.
8a2-4a+6a-3=0
8a2-4a+6a-3=0
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(8a2-4a)+6a-3=0
Factor out the greatest common factor (GCF) from each group.
4a(2a-1)+3(2a-1)=0
4a(2a-1)+3(2a-1)=0
Factor the polynomial by factoring out the greatest common factor, 2a-1.
(2a-1)(4a+3)=0
(2a-1)(4a+3)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
2a-1=0
4a+3=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
2a-1=0
Add 1 to both sides of the equation.
2a=1
Divide each term by 2 and simplify.
Divide each term in 2a=1 by 2.
2a2=12
Cancel the common factor of 2.
Cancel the common factor.
2a2=12
Divide a by 1.
a=12
a=12
a=12
a=12
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
4a+3=0
Subtract 3 from both sides of the equation.
4a=-3
Divide each term by 4 and simplify.
Divide each term in 4a=-3 by 4.
4a4=-34
Cancel the common factor of 4.
Cancel the common factor.
4a4=-34
Divide a by 1.
a=-34
a=-34
Move the negative in front of the fraction.
a=-34
a=-34
a=-34
The final solution is all the values that make (2a-1)(4a+3)=0 true.
a=12,-34
a=12,-34
The result can be shown in multiple forms.
Exact Form:
a=12,-34
Decimal Form:
a=0.5,-0.75
Solve for a 1/3*(4a+1)=1/(2a)

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