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Subtracting 3b2−4b−5−2b2−6b+5\frac{3}{b^2-4b-5} - \frac{2}{b^2-6b+5}

Subtract two rational expressions with different factorable trinomial denominators:

3b2−4b−5−2b2−6b+5\frac{3}{b^2 - 4b - 5} - \frac{2}{b^2 - 6b + 5}

Step 1 — Factor the denominators to find the LCD. Factor each trinomial: b2−4b−5=(b−5)(b+1)b^2 - 4b - 5 = (b - 5)(b + 1) b2−6b+5=(b−5)(b−1)b^2 - 6b + 5 = (b - 5)(b - 1) The LCD is (b−5)(b+1)(b−1)(b - 5)(b + 1)(b - 1).

Step 2 — Rewrite each fraction with the LCD. Multiply each fraction by its missing factor: 3(b−1)(b−5)(b+1)(b−1)−2(b+1)(b−5)(b−1)(b+1)\frac{3(b - 1)}{(b - 5)(b + 1)(b - 1)} - \frac{2(b + 1)}{(b - 5)(b - 1)(b + 1)}

Distribute the numerators: 3b−3(b−5)(b+1)(b−1)−2b+2(b−5)(b+1)(b−1)\frac{3b - 3}{(b - 5)(b + 1)(b - 1)} - \frac{2b + 2}{(b - 5)(b + 1)(b - 1)}

Step 3 — Subtract the numerators. Place the second numerator in parentheses and distribute the negative sign: 3b−3−(2b+2)(b−5)(b+1)(b−1)\frac{3b - 3 - (2b + 2)}{(b - 5)(b + 1)(b - 1)}

3b−3−2b−2(b−5)(b+1)(b−1)\frac{3b - 3 - 2b - 2}{(b - 5)(b + 1)(b - 1)}

Combine like terms: b−5(b−5)(b+1)(b−1)\frac{b - 5}{(b - 5)(b + 1)(b - 1)}

Step 4 — Simplify. Cancel the shared factor (b−5)(b - 5): 1(b+1)(b−1)\frac{1}{(b + 1)(b - 1)}

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Updated 2026-04-30

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References

Tags

OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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