Solve for k 2+5/(3k-6)=-2/((3k-6)^2)

Math
2+53k-6=-2(3k-6)2
Move all terms not containing k to the right side of the equation.
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Subtract 2 from both sides of the equation.
53k-6=-2(3k-6)2-2
Simplify each term.
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Simplify the denominator.
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Factor 3 out of 3k-6.
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Factor 3 out of 3k.
53k-6=-2(3(k)-6)2-2
Factor 3 out of -6.
53k-6=-2(3k+3⋅-2)2-2
Factor 3 out of 3k+3⋅-2.
53k-6=-2(3(k-2))2-2
53k-6=-2(3(k-2))2-2
Apply the product rule to 3(k-2).
53k-6=-232(k-2)2-2
Raise 3 to the power of 2.
53k-6=-29(k-2)2-2
53k-6=-29(k-2)2-2
Move the negative in front of the fraction.
53k-6=-29(k-2)2-2
53k-6=-29(k-2)2-2
53k-6=-29(k-2)2-2
Factor 3 out of 3k-6.
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Factor 3 out of 3k.
53(k)-6=-29(k-2)2-2
Factor 3 out of -6.
53k+3⋅-2=-29(k-2)2-2
Factor 3 out of 3k+3⋅-2.
53(k-2)=-29(k-2)2-2
53(k-2)=-29(k-2)2-2
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
3(k-2),9(k-2)2,1
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.
Since 3 has no factors besides 1 and 3.
3 is a prime number
9 has factors of 3 and 3.
3⋅3
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 3,9,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
3⋅3
Multiply 3 by 3.
9
The factor for k-2 is k-2 itself.
(k-2)=k-2
(k-2) occurs 1 time.
The factors for k-2 are (k-2)⋅(k-2), which is k-2 multiplied by itself 2 times.
(k-2)=(k-2)⋅(k-2)
(k-2) occurs 2 times.
The LCM of k-2,(k-2)2 is the result of multiplying all factors the greatest number of times they occur in either term.
(k-2)2
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
9(k-2)2
9(k-2)2
Multiply each term by 9(k-2)2 and simplify.
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Multiply each term in 53(k-2)=-29(k-2)2-2 by 9(k-2)2 in order to remove all the denominators from the equation.
53(k-2)⋅(9(k-2)2)=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Simplify 53(k-2)⋅(9(k-2)2).
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Rewrite using the commutative property of multiplication.
953(k-2)(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Cancel the common factor of 3.
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Factor 3 out of 9.
3(3)53(k-2)(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Cancel the common factor.
3⋅353(k-2)(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Rewrite the expression.
35k-2(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
35k-2(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Combine 3 and 5k-2.
3⋅5k-2(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Multiply 3 by 5.
15k-2(k-2)2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Cancel the common factor of k-2.
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Factor k-2 out of (k-2)2.
15k-2((k-2)(k-2))=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Cancel the common factor.
15k-2((k-2)(k-2))=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Rewrite the expression.
15(k-2)=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
15(k-2)=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Apply the distributive property.
15k+15⋅-2=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Multiply 15 by -2.
15k-30=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
15k-30=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Simplify -29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2).
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Simplify each term.
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Cancel the common factor of 9(k-2)2.
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Move the leading negative in -29(k-2)2 into the numerator.
15k-30=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Cancel the common factor.
15k-30=-29(k-2)2⋅(9(k-2)2)-2⋅(9(k-2)2)
Rewrite the expression.
15k-30=-2-2⋅(9(k-2)2)
15k-30=-2-2⋅(9(k-2)2)
Rewrite (k-2)2 as (k-2)(k-2).
15k-30=-2-2⋅(9((k-2)(k-2)))
Expand (k-2)(k-2) using the FOIL Method.
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Apply the distributive property.
15k-30=-2-2⋅(9(k(k-2)-2(k-2)))
Apply the distributive property.
15k-30=-2-2⋅(9(k⋅k+k⋅-2-2(k-2)))
Apply the distributive property.
15k-30=-2-2⋅(9(k⋅k+k⋅-2-2k-2⋅-2))
15k-30=-2-2⋅(9(k⋅k+k⋅-2-2k-2⋅-2))
Simplify and combine like terms.
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Simplify each term.
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Multiply k by k.
15k-30=-2-2⋅(9(k2+k⋅-2-2k-2⋅-2))
Move -2 to the left of k.
15k-30=-2-2⋅(9(k2-2⋅k-2k-2⋅-2))
Multiply -2 by -2.
15k-30=-2-2⋅(9(k2-2k-2k+4))
15k-30=-2-2⋅(9(k2-2k-2k+4))
Subtract 2k from -2k.
15k-30=-2-2⋅(9(k2-4k+4))
15k-30=-2-2⋅(9(k2-4k+4))
Apply the distributive property.
15k-30=-2-2⋅(9k2+9(-4k)+9⋅4)
Simplify.
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Multiply -4 by 9.
15k-30=-2-2⋅(9k2-36k+9⋅4)
Multiply 9 by 4.
15k-30=-2-2⋅(9k2-36k+36)
15k-30=-2-2⋅(9k2-36k+36)
Apply the distributive property.
15k-30=-2-2(9k2)-2(-36k)-2⋅36
Simplify.
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Multiply 9 by -2.
15k-30=-2-18k2-2(-36k)-2⋅36
Multiply -36 by -2.
15k-30=-2-18k2+72k-2⋅36
Multiply -2 by 36.
15k-30=-2-18k2+72k-72
15k-30=-2-18k2+72k-72
15k-30=-2-18k2+72k-72
Subtract 72 from -2.
15k-30=-18k2+72k-74
15k-30=-18k2+72k-74
15k-30=-18k2+72k-74
Solve the equation.
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Since k is on the right side of the equation, switch the sides so it is on the left side of the equation.
-18k2+72k-74=15k-30
Move all terms containing k to the left side of the equation.
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Subtract 15k from both sides of the equation.
-18k2+72k-74-15k=-30
Subtract 15k from 72k.
-18k2+57k-74=-30
-18k2+57k-74=-30
Move 30 to the left side of the equation by adding it to both sides.
-18k2+57k-74+30=0
Add -74 and 30.
-18k2+57k-44=0
Factor the left side of the equation.
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Factor -1 out of -18k2+57k-44.
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Factor -1 out of -18k2.
-(18k2)+57k-44=0
Factor -1 out of 57k.
-(18k2)-(-57k)-44=0
Rewrite -44 as -1(44).
-(18k2)-(-57k)-1⋅44=0
Factor -1 out of -(18k2)-(-57k).
-(18k2-57k)-1⋅44=0
Factor -1 out of -(18k2-57k)-1(44).
-(18k2-57k+44)=0
-(18k2-57k+44)=0
Factor.
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Factor by grouping.
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For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=18⋅44=792 and whose sum is b=-57.
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Factor -57 out of -57k.
-(18k2-57k+44)=0
Rewrite -57 as -24 plus -33
-(18k2+(-24-33)k+44)=0
Apply the distributive property.
-(18k2-24k-33k+44)=0
-(18k2-24k-33k+44)=0
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
-((18k2-24k)-33k+44)=0
Factor out the greatest common factor (GCF) from each group.
-(6k(3k-4)-11(3k-4))=0
-(6k(3k-4)-11(3k-4))=0
Factor the polynomial by factoring out the greatest common factor, 3k-4.
-((3k-4)(6k-11))=0
-((3k-4)(6k-11))=0
Remove unnecessary parentheses.
-(3k-4)(6k-11)=0
-(3k-4)(6k-11)=0
-(3k-4)(6k-11)=0
Multiply each term in -(3k-4)(6k-11)=0 by -1
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Multiply each term in -(3k-4)(6k-11)=0 by -1.
(-(3k-4)(6k-11))⋅-1=0⋅-1
Simplify (-(3k-4)(6k-11))⋅-1.
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Simplify by multiplying through.
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Apply the distributive property.
(-(3k)–4)(6k-11)⋅-1=0⋅-1
Multiply.
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Multiply 3 by -1.
(-3k–4)(6k-11)⋅-1=0⋅-1
Multiply -1 by -4.
(-3k+4)(6k-11)⋅-1=0⋅-1
(-3k+4)(6k-11)⋅-1=0⋅-1
(-3k+4)(6k-11)⋅-1=0⋅-1
Expand (-3k+4)(6k-11) using the FOIL Method.
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Apply the distributive property.
(-3k(6k-11)+4(6k-11))⋅-1=0⋅-1
Apply the distributive property.
(-3k(6k)-3k⋅-11+4(6k-11))⋅-1=0⋅-1
Apply the distributive property.
(-3k(6k)-3k⋅-11+4(6k)+4⋅-11)⋅-1=0⋅-1
(-3k(6k)-3k⋅-11+4(6k)+4⋅-11)⋅-1=0⋅-1
Simplify and combine like terms.
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Simplify each term.
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Multiply k by k.
(-3⋅6k2-3k⋅-11+4(6k)+4⋅-11)⋅-1=0⋅-1
Multiply -3 by 6.
(-18k2-3k⋅-11+4(6k)+4⋅-11)⋅-1=0⋅-1
Multiply -11 by -3.
(-18k2+33k+4(6k)+4⋅-11)⋅-1=0⋅-1
Multiply 6 by 4.
(-18k2+33k+24k+4⋅-11)⋅-1=0⋅-1
Multiply 4 by -11.
(-18k2+33k+24k-44)⋅-1=0⋅-1
(-18k2+33k+24k-44)⋅-1=0⋅-1
Add 33k and 24k.
(-18k2+57k-44)⋅-1=0⋅-1
(-18k2+57k-44)⋅-1=0⋅-1
Apply the distributive property.
-18k2⋅-1+57k⋅-1-44⋅-1=0⋅-1
Simplify.
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Multiply -1 by -18.
18k2+57k⋅-1-44⋅-1=0⋅-1
Multiply -1 by 57.
18k2-57k-44⋅-1=0⋅-1
Multiply -44 by -1.
18k2-57k+44=0⋅-1
18k2-57k+44=0⋅-1
18k2-57k+44=0⋅-1
Multiply 0 by -1.
18k2-57k+44=0
18k2-57k+44=0
Factor by grouping.
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For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=18⋅44=792 and whose sum is b=-57.
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Factor -57 out of -57k.
18k2-57k+44=0
Rewrite -57 as -24 plus -33
18k2+(-24-33)k+44=0
Apply the distributive property.
18k2-24k-33k+44=0
18k2-24k-33k+44=0
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
(18k2-24k)-33k+44=0
Factor out the greatest common factor (GCF) from each group.
6k(3k-4)-11(3k-4)=0
6k(3k-4)-11(3k-4)=0
Factor the polynomial by factoring out the greatest common factor, 3k-4.
(3k-4)(6k-11)=0
(3k-4)(6k-11)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
3k-4=0
6k-11=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
3k-4=0
Add 4 to both sides of the equation.
3k=4
Divide each term by 3 and simplify.
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Divide each term in 3k=4 by 3.
3k3=43
Cancel the common factor of 3.
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Cancel the common factor.
3k3=43
Divide k by 1.
k=43
k=43
k=43
k=43
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
6k-11=0
Add 11 to both sides of the equation.
6k=11
Divide each term by 6 and simplify.
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Divide each term in 6k=11 by 6.
6k6=116
Cancel the common factor of 6.
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Cancel the common factor.
6k6=116
Divide k by 1.
k=116
k=116
k=116
k=116
The final solution is all the values that make (3k-4)(6k-11)=0 true.
k=43,116
k=43,116
The result can be shown in multiple forms.
Exact Form:
k=43,116
Decimal Form:
k=1.3‾,1.83‾
Mixed Number Form:
k=113,156
Solve for k 2+5/(3k-6)=-2/((3k-6)^2)

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