Mister Exam
How do you k/(k+6)^2-k/(k^2-36)+1/(k-6)^2 in partial fractions?
The solution
You have entered
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k k 1
-------- - ------- + --------
2 2 2
(k + 6) k - 36 (k - 6)
$$\left(- \frac{k}{k^{2} - 36} + \frac{k}{\left(k + 6\right)^{2}}\right) + \frac{1}{\left(k - 6\right)^{2}}$$
k/(k + 6)^2 - k/(k^2 - 36) + 1/((k - 6)^2)
Fraction decomposition
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(-6 + k)^(-2) + 1/(2*(6 + k)) - 6/(6 + k)^2 - 1/(2*(-6 + k))
$$\frac{1}{2 \left(k + 6\right)} - \frac{6}{\left(k + 6\right)^{2}} - \frac{1}{2 \left(k - 6\right)} + \frac{1}{\left(k - 6\right)^{2}}$$
1 1 6 1
--------- + --------- - -------- - ----------
2 2*(6 + k) 2 2*(-6 + k)
(-6 + k) (6 + k)
General simplification
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2
36 - 11*k + 84*k
-----------------
4 2
1296 + k - 72*k
$$\frac{- 11 k^{2} + 84 k + 36}{k^{4} - 72 k^{2} + 1296}$$
(36 - 11*k^2 + 84*k)/(1296 + k^4 - 72*k^2)
Combinatorics
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/ 2\
-\-36 - 84*k + 11*k /
----------------------
2 2
(-6 + k) *(6 + k)
$$- \frac{11 k^{2} - 84 k - 36}{\left(k - 6\right)^{2} \left(k + 6\right)^{2}}$$
-(-36 - 84*k + 11*k^2)/((-6 + k)^2*(6 + k)^2)
Assemble expression
[src]
1 k k
--------- + -------- - --------
2 2 2
(-6 + k) (6 + k) -36 + k
$$- \frac{k}{k^{2} - 36} + \frac{k}{\left(k + 6\right)^{2}} + \frac{1}{\left(k - 6\right)^{2}}$$
(-6 + k)^(-2) + k/(6 + k)^2 - k/(-36 + k^2)
Powers
[src]
1 k k
--------- + -------- - --------
2 2 2
(-6 + k) (6 + k) -36 + k
$$- \frac{k}{k^{2} - 36} + \frac{k}{\left(k + 6\right)^{2}} + \frac{1}{\left(k - 6\right)^{2}}$$
(-6 + k)^(-2) + k/(6 + k)^2 - k/(-36 + k^2)
Trigonometric part
[src]
1 k k
--------- + -------- - --------
2 2 2
(-6 + k) (6 + k) -36 + k
$$- \frac{k}{k^{2} - 36} + \frac{k}{\left(k + 6\right)^{2}} + \frac{1}{\left(k - 6\right)^{2}}$$
(-6 + k)^(-2) + k/(6 + k)^2 - k/(-36 + k^2)
Common denominator
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/ 2\
-\-36 - 84*k + 11*k /
----------------------
4 2
1296 + k - 72*k
$$- \frac{11 k^{2} - 84 k - 36}{k^{4} - 72 k^{2} + 1296}$$
-(-36 - 84*k + 11*k^2)/(1296 + k^4 - 72*k^2)
Rational denominator
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2 / / 2\ 2\ 2 / 2\
(-6 + k) *\k*\-36 + k / - k*(6 + k) / + (6 + k) *\-36 + k /
-----------------------------------------------------------
/ 2\ 2 2
\-36 + k /*(-6 + k) *(6 + k)
$$\frac{\left(k - 6\right)^{2} \left(- k \left(k + 6\right)^{2} + k \left(k^{2} - 36\right)\right) + \left(k + 6\right)^{2} \left(k^{2} - 36\right)}{\left(k - 6\right)^{2} \left(k + 6\right)^{2} \left(k^{2} - 36\right)}$$
((-6 + k)^2*(k*(-36 + k^2) - k*(6 + k)^2) + (6 + k)^2*(-36 + k^2))/((-36 + k^2)*(-6 + k)^2*(6 + k)^2)
Numerical answer
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0.0277777777777778/(-1 + 0.166666666666667*k)^2 - k/(-36.0 + k^2) + 0.0277777777777778*k/(1 + 0.166666666666667*k)^2
0.0277777777777778/(-1 + 0.166666666666667*k)^2 - k/(-36.0 + k^2) + 0.0277777777777778*k/(1 + 0.166666666666667*k)^2
Combining rational expressions
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2 / 2\ 2 / 2 2\
(6 + k) *\-36 + k / + k*(-6 + k) *\-36 + k - (6 + k) /
-------------------------------------------------------
/ 2\ 2 2
\-36 + k /*(-6 + k) *(6 + k)
$$\frac{k \left(k - 6\right)^{2} \left(k^{2} - \left(k + 6\right)^{2} - 36\right) + \left(k + 6\right)^{2} \left(k^{2} - 36\right)}{\left(k - 6\right)^{2} \left(k + 6\right)^{2} \left(k^{2} - 36\right)}$$
((6 + k)^2*(-36 + k^2) + k*(-6 + k)^2*(-36 + k^2 - (6 + k)^2))/((-36 + k^2)*(-6 + k)^2*(6 + k)^2)