Mister Exam
How do you (3*y^2-12)/(2*y^2-15*y+18) in partial fractions?
The solution
You have entered
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2 3*y - 12 ---------------- 2 2*y - 15*y + 18
$$\frac{3 y^{2} - 12}{\left(2 y^{2} - 15 y\right) + 18}$$
(3*y^2 - 12)/(2*y^2 - 15*y + 18)
General simplification
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/ 2\
3*\-4 + y /
----------------
2
18 - 15*y + 2*y
$$\frac{3 \left(y^{2} - 4\right)}{2 y^{2} - 15 y + 18}$$
3*(-4 + y^2)/(18 - 15*y + 2*y^2)
Fraction decomposition
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3/2 + 7/(6*(-3 + 2*y)) + 32/(3*(-6 + y))
$$\frac{3}{2} + \frac{7}{6 \left(2 y - 3\right)} + \frac{32}{3 \left(y - 6\right)}$$
3 7 32 - + ------------ + ---------- 2 6*(-3 + 2*y) 3*(-6 + y)
Numerical answer
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(-12.0 + 3.0*y^2)/(18.0 + 2.0*y^2 - 15.0*y)
(-12.0 + 3.0*y^2)/(18.0 + 2.0*y^2 - 15.0*y)
Rational denominator
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2
-12 + 3*y
----------------
2
18 - 15*y + 2*y
$$\frac{3 y^{2} - 12}{2 y^{2} - 15 y + 18}$$
(-12 + 3*y^2)/(18 - 15*y + 2*y^2)
Assemble expression
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2
-12 + 3*y
----------------
2
18 - 15*y + 2*y
$$\frac{3 y^{2} - 12}{2 y^{2} - 15 y + 18}$$
(-12 + 3*y^2)/(18 - 15*y + 2*y^2)
Trigonometric part
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2
-12 + 3*y
----------------
2
18 - 15*y + 2*y
$$\frac{3 y^{2} - 12}{2 y^{2} - 15 y + 18}$$
(-12 + 3*y^2)/(18 - 15*y + 2*y^2)
Common denominator
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3 -78 + 45*y
- + ----------------
2 2
36 - 30*y + 4*y
$$\frac{45 y - 78}{4 y^{2} - 30 y + 36} + \frac{3}{2}$$
3/2 + (-78 + 45*y)/(36 - 30*y + 4*y^2)
Powers
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2
-12 + 3*y
----------------
2
18 - 15*y + 2*y
$$\frac{3 y^{2} - 12}{2 y^{2} - 15 y + 18}$$
(-12 + 3*y^2)/(18 - 15*y + 2*y^2)
Combining rational expressions
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/ 2\ 3*\-4 + y / ------------------ 18 + y*(-15 + 2*y)
$$\frac{3 \left(y^{2} - 4\right)}{y \left(2 y - 15\right) + 18}$$
3*(-4 + y^2)/(18 + y*(-15 + 2*y))
Combinatorics
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3*(-2 + y)*(2 + y) ------------------- (-6 + y)*(-3 + 2*y)
$$\frac{3 \left(y - 2\right) \left(y + 2\right)}{\left(y - 6\right) \left(2 y - 3\right)}$$
3*(-2 + y)*(2 + y)/((-6 + y)*(-3 + 2*y))