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Question: factor 15n²-27n-6

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To factor the quadratic expression $$15n² - 27n - 6$$, we need to find two binomials whose product gives the original quadratic expression. We start with polynomial factorization techniques. First, recognize the quadratic in its general form: $$Ax^2 + Bx + C$$. For $$15n² - 27n - 6$$: 1. **Identify A, B, and C:** - A (coefficient of $$n²$$) = 15 - B (coefficient of $$n$$) = -27 - C (constant term) = -6 2. **Find two numbers that multiply to $$A \cdot C$$ and add to $$B$$:** - $$A \cdot C = 15 \times -6 = -90$$ - We need to find two numbers that multiply to -90 and add to -27. After checking possible pairs, we find that -30 and +3 work: - (-30) + 3 = -27 - (-30) * 3 = -90 3. **Rewrite the middle term using the found pair:** $$15n² - 27n - 6$$ can be rewritten as $$15n² - 30n + 3n - 6$$. 4. **Factor by grouping:** Group the terms to factor out the common factors from each group: - $$15n² - 30n + 3n - 6$$ - Group into two pairs: $$(15n² - 30n) + (3n - 6)$$ 5. **Factor out the greatest common factor from each pair:** - $$15n(n - 2) + 3(n - 2)$$ 6. **Factor out the common binomial factor:** - Because both terms contain $$(n - 2)$$, factor this out: - $$(15n + 3)(n - 2)$$ 7. **Simplify the expression:** - Notice that $$15n + 3$$ has a common factor of 3: - $$3(5n + 1)(n - 2)$$ So, the factored form of the quadratic expression $$15n² - 27n - 6$$ is: $3(5n + 1)(n - 2)$

May 24, 2024, 5:50 a.m.

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