Partial Fractions: (2x + 3)/((x + 1)(x² + 4))
(2x + 3)/((x + 1)(x² + 4)) decomposes into (1/5)/(x + 1) + (-(1/5)x + 11/5)/(x^2 + 4). A worked irreducible quadratic example with the factorisation, the coefficient system, and an editable calculator.
(2x + 3)/((x + 1)(x² + 4)) = (1/5)/(x + 1) + (-(1/5)x + 11/5)/(x^2 + 4). One linear factor and one irreducible quadratic — the usual setup in Laplace problems.
e.g. 3x+11
Expanded or factored, e.g. (x-2)(x+3) or x^2+x-6
The decomposition
(2x + 3)/((x + 1)(x² + 4)) = (1/5)/(x + 1) + (-(1/5)x + 11/5)/(x^2 + 4)
Denominator factors: (x + 1) × (x^2 + 4)
This is a irreducible quadratic problem. One linear factor and one irreducible quadratic — the usual setup in Laplace problems.
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Frequently asked questions
How does (2x + 3)/((x + 1)(x² + 4)) decompose? ▾
(2x + 3)/((x + 1)(x² + 4)) = (1/5)/(x + 1) + (-(1/5)x + 11/5)/(x^2 + 4). One linear factor and one irreducible quadratic — the usual setup in Laplace problems.
How was the denominator factored? ▾
It factors as (x + 1) × (x^2 + 4), which sets the shape of the decomposition.
How do I check the answer? ▾
Add the resulting fractions back over a common denominator — you should recover the original expression. Substituting a convenient value of x into both sides is a quicker spot check.
About this Calculator
(2x + 3)/((x + 1)(x² + 4)) decomposes into (1/5)/(x + 1) + (-(1/5)x + 11/5)/(x^2 + 4). A worked irreducible quadratic example with the factorisation, the coefficient system, and an editable calculator.