Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
-10-7*((x-2)/2-3/(x-2))+18/(x-2)^2+(x-2)^2/2
How do you -10-7*((x-2)/2-3/(x-2))+18/(x-2)^2+(x-2)^2/2 in partial fractions?
The solution
You have entered
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2
/x - 2 3 \ 18 (x - 2)
-10 - 7*|----- - -----| + -------- + --------
\ 2 x - 2/ 2 2
(x - 2)
$$\frac{\left(x - 2\right)^{2}}{2} + \left(\left(- 7 \left(\frac{x - 2}{2} - \frac{3}{x - 2}\right) - 10\right) + \frac{18}{\left(x - 2\right)^{2}}\right)$$
-10 - 7*((x - 2)/2 - 3/(x - 2)) + 18/(x - 2)^2 + (x - 2)^2/2
General simplification
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4 3 2
-56 + x - 15*x + 6*x + 46*x
------------------------------
/ 2 \
2*\4 + x - 4*x/
$$\frac{x^{4} - 15 x^{3} + 46 x^{2} + 6 x - 56}{2 \left(x^{2} - 4 x + 4\right)}$$
(-56 + x^4 - 15*x^3 + 6*x + 46*x^2)/(2*(4 + x^2 - 4*x))
Fraction decomposition
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-1 + x^2/2 + 18/(-2 + x)^2 + 21/(-2 + x) - 11*x/2
$$\frac{x^{2}}{2} - \frac{11 x}{2} - 1 + \frac{21}{x - 2} + \frac{18}{\left(x - 2\right)^{2}}$$
2
x 18 21 11*x
-1 + -- + --------- + ------ - ----
2 2 -2 + x 2
(-2 + x)
Trigonometric part
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2
(-2 + x) 18 21 7*x
-3 + --------- + --------- + ------ - ---
2 2 -2 + x 2
(-2 + x)
$$- \frac{7 x}{2} + \frac{\left(x - 2\right)^{2}}{2} - 3 + \frac{21}{x - 2} + \frac{18}{\left(x - 2\right)^{2}}$$
-3 + (-2 + x)^2/2 + 18/(-2 + x)^2 + 21/(-2 + x) - 7*x/2
Numerical answer
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-3.0 + 2.0*(-1 + 0.5*x)^2 + 4.5/(-1 + 0.5*x)^2 + 21.0/(-2.0 + x) - 3.5*x
-3.0 + 2.0*(-1 + 0.5*x)^2 + 4.5/(-1 + 0.5*x)^2 + 21.0/(-2.0 + x) - 3.5*x
Powers
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2
(-2 + x) 18 21 7*x
-3 + --------- + --------- + ------ - ---
2 2 -2 + x 2
(-2 + x)
$$- \frac{7 x}{2} + \frac{\left(x - 2\right)^{2}}{2} - 3 + \frac{21}{x - 2} + \frac{18}{\left(x - 2\right)^{2}}$$
-3 + (-2 + x)^2/2 + 18/(-2 + x)^2 + 21/(-2 + x) - 7*x/2
Rational denominator
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4 2 / 2\
-144 + 72*x + (-2 + x) *(-4 + 2*x) + 2*(-2 + x) *\82 - 20*x - 7*(-2 + x) /
--------------------------------------------------------------------------
2
2*(-4 + 2*x)*(-2 + x)
$$\frac{72 x + \left(x - 2\right)^{4} \left(2 x - 4\right) + 2 \left(x - 2\right)^{2} \left(- 20 x - 7 \left(x - 2\right)^{2} + 82\right) - 144}{2 \left(x - 2\right)^{2} \left(2 x - 4\right)}$$
(-144 + 72*x + (-2 + x)^4*(-4 + 2*x) + 2*(-2 + x)^2*(82 - 20*x - 7*(-2 + x)^2))/(2*(-4 + 2*x)*(-2 + x)^2)
Combinatorics
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/ 2 \
(1 + x)*(-4 + x)*\14 + x - 12*x/
---------------------------------
2
2*(-2 + x)
$$\frac{\left(x - 4\right) \left(x + 1\right) \left(x^{2} - 12 x + 14\right)}{2 \left(x - 2\right)^{2}}$$
(1 + x)*(-4 + x)*(14 + x^2 - 12*x)/(2*(-2 + x)^2)
Combining rational expressions
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4 / 2\
36 + (-2 + x) + (-2 + x)*\82 - 20*x - 7*(-2 + x) /
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2
2*(-2 + x)
$$\frac{\left(x - 2\right)^{4} + \left(x - 2\right) \left(- 20 x - 7 \left(x - 2\right)^{2} + 82\right) + 36}{2 \left(x - 2\right)^{2}}$$
(36 + (-2 + x)^4 + (-2 + x)*(82 - 20*x - 7*(-2 + x)^2))/(2*(-2 + x)^2)
Assemble expression
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2
(-2 + x) 18 21 7*x
-3 + --------- + --------- + ------ - ---
2 2 -2 + x 2
(-2 + x)
$$- \frac{7 x}{2} + \frac{\left(x - 2\right)^{2}}{2} - 3 + \frac{21}{x - 2} + \frac{18}{\left(x - 2\right)^{2}}$$
-3 + (-2 + x)^2/2 + 18/(-2 + x)^2 + 21/(-2 + x) - 7*x/2
Common denominator
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2
x 11*x -24 + 21*x
-1 + -- - ---- + ------------
2 2 2
4 + x - 4*x
$$\frac{x^{2}}{2} - \frac{11 x}{2} + \frac{21 x - 24}{x^{2} - 4 x + 4} - 1$$
-1 + x^2/2 - 11*x/2 + (-24 + 21*x)/(4 + x^2 - 4*x)