# Solve for t 8/(t^2-16)-1/8=1/(t-4)

8t2-16-18=1t-4
Add 18 to both sides of the equation.
8t2-16=1t-4+18
Factor each term.
Rewrite 16 as 42.
8t2-42=1t-4+18
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=t and b=4.
8(t+4)(t-4)=1t-4+18
8(t+4)(t-4)=1t-4+18
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.
(t+4)(t-4),t-4,8
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 prime factors for 8 are 2⋅2⋅2.
8 has factors of 2 and 4.
2⋅4
4 has factors of 2 and 2.
2⋅2⋅2
2⋅2⋅2
The LCM of 1,1,8 is 2⋅2⋅2=8.
Multiply 2 by 2.
4⋅2
Multiply 4 by 2.
8
8
The factor for t+4 is t+4 itself.
(t+4)=t+4
(t+4) occurs 1 time.
The factor for t-4 is t-4 itself.
(t-4)=t-4
(t-4) occurs 1 time.
The LCM of t+4,t-4,t-4 is the result of multiplying all factors the greatest number of times they occur in either term.
(t+4)(t-4)
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
8(t+4)(t-4)
8(t+4)(t-4)
Multiply each term by 8(t+4)(t-4) and simplify.
Multiply each term in 8(t+4)(t-4)=1t-4+18 by 8(t+4)(t-4) in order to remove all the denominators from the equation.
8(t+4)(t-4)⋅(8(t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Simplify 8(t+4)(t-4)⋅(8(t+4)(t-4)).
Rewrite using the commutative property of multiplication.
88(t+4)(t-4)((t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Multiply 88(t+4)(t-4).
Combine 8 and 8(t+4)(t-4).
8⋅8(t+4)(t-4)((t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Multiply 8 by 8.
64(t+4)(t-4)((t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
64(t+4)(t-4)((t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Cancel the common factor of (t+4)(t-4).
Cancel the common factor.
64(t+4)(t-4)((t+4)(t-4))=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Rewrite the expression.
64=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
64=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
64=1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4))
Simplify 1t-4⋅(8(t+4)(t-4))+18⋅(8(t+4)(t-4)).
Simplify each term.
Rewrite using the commutative property of multiplication.
64=81t-4((t+4)(t-4))+18⋅(8(t+4)(t-4))
Combine 8 and 1t-4.
64=8t-4((t+4)(t-4))+18⋅(8(t+4)(t-4))
Cancel the common factor of t-4.
Factor t-4 out of (t+4)(t-4).
64=8t-4((t-4)(t+4))+18⋅(8(t+4)(t-4))
Cancel the common factor.
64=8t-4((t-4)(t+4))+18⋅(8(t+4)(t-4))
Rewrite the expression.
64=8(t+4)+18⋅(8(t+4)(t-4))
64=8(t+4)+18⋅(8(t+4)(t-4))
Apply the distributive property.
64=8t+8⋅4+18⋅(8(t+4)(t-4))
Multiply 8 by 4.
64=8t+32+18⋅(8(t+4)(t-4))
Cancel the common factor of 8.
Factor 8 out of 8(t+4)(t-4).
64=8t+32+18⋅(8((t+4)(t-4)))
Cancel the common factor.
64=8t+32+18⋅(8((t+4)(t-4)))
Rewrite the expression.
64=8t+32+(t+4)(t-4)
64=8t+32+(t+4)(t-4)
Expand (t+4)(t-4) using the FOIL Method.
Apply the distributive property.
64=8t+32+t(t-4)+4(t-4)
Apply the distributive property.
64=8t+32+t⋅t+t⋅-4+4(t-4)
Apply the distributive property.
64=8t+32+t⋅t+t⋅-4+4t+4⋅-4
64=8t+32+t⋅t+t⋅-4+4t+4⋅-4
Combine the opposite terms in t⋅t+t⋅-4+4t+4⋅-4.
Reorder the factors in the terms t⋅-4 and 4t.
64=8t+32+t⋅t-4t+4t+4⋅-4
Add -4t and 4t.
64=8t+32+t⋅t+0+4⋅-4
Add t⋅t and 0.
64=8t+32+t⋅t+4⋅-4
64=8t+32+t⋅t+4⋅-4
Simplify each term.
Multiply t by t.
64=8t+32+t2+4⋅-4
Multiply 4 by -4.
64=8t+32+t2-16
64=8t+32+t2-16
64=8t+32+t2-16
Subtract 16 from 32.
64=8t+t2+16
64=8t+t2+16
64=8t+t2+16
Solve the equation.
Rewrite the equation as 8t+t2+16=64.
8t+t2+16=64
Set the equation equal to zero.
Move 64 to the left side of the equation by subtracting it from both sides.
8t+t2+16-64=0
Subtract 64 from 16.
8t+t2-48=0
8t+t2-48=0
Factor the left side of the equation.
Let u=t. Substitute u for all occurrences of t.
8u+u2-48
Factor 8u+u2-48 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 -48 and whose sum is 8.
-4,12
Write the factored form using these integers.
(u-4)(u+12)
(u-4)(u+12)
Replace all occurrences of u with t.
(t-4)(t+12)
Replace the left side with the factored expression.
(t-4)(t+12)=0
(t-4)(t+12)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
t-4=0
t+12=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
t-4=0
Add 4 to both sides of the equation.
t=4
t=4
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
t+12=0
Subtract 12 from both sides of the equation.
t=-12
t=-12
The final solution is all the values that make (t-4)(t+12)=0 true.
t=4,-12
t=4,-12
Exclude the solutions that do not make 8t2-16-18=1t-4 true.
t=-12
Solve for t 8/(t^2-16)-1/8=1/(t-4)

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