Factor completely: 2w² + 11w + 9
Understand the Problem
The question is asking for the complete factorization of the quadratic expression given as 2w² + 11w + 9. To solve it, we will need to identify two binomials that multiply to give this quadratic expression.
Answer
$$(2w + 9)(w + 1)$$
Answer for screen readers
The complete factorization of the quadratic expression (2w^2 + 11w + 9) is: $$(2w + 9)(w + 1)$$
Steps to Solve
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Identify the quadratic expression The given quadratic expression is (2w^2 + 11w + 9).
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Multiply leading coefficient by constant term Multiply the leading coefficient (2) by the constant term (9): $$ 2 \times 9 = 18 $$
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Find two numbers that multiply to 18 and add to 11 We need to find two numbers that multiply to 18 and add to 11. The numbers are 9 and 2, since: $$ 9 \times 2 = 18 $$ $$ 9 + 2 = 11 $$
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Rewrite the middle term using the found numbers Rewrite (11w) as (9w + 2w): $$ 2w^2 + 9w + 2w + 9 $$
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Group the terms Group the first two terms and the last two terms: $$ (2w^2 + 9w) + (2w + 9) $$
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Factor by grouping Factor out the common terms in each group: $$ w(2w + 9) + 1(2w + 9) $$
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Factor out the common binomial factor Now factor out the common binomial ((2w + 9)): $$ (2w + 9)(w + 1) $$
The complete factorization of the quadratic expression (2w^2 + 11w + 9) is: $$(2w + 9)(w + 1)$$
More Information
The quadratic expression is factored into two binomials that can be verified by expanding them back to the original expression. This is a common technique in algebra to simplify expressions.
Tips
- Forgetting to account for the leading coefficient when identifying number pairs.
- Miscalculating the pairs of numbers that multiply to the constant term.
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