Factor completely: 2w² + 11w + 9.
Understand the Problem
The question is asking to factor the quadratic expression completely. Specifically, it needs to be done for the expression 2w² + 11w + 9.
Answer
The expression $2w^2 + 11w + 9$ factors to $(2w + 9)(w + 1)$.
Answer for screen readers
The completely factored form of the expression is $(2w + 9)(w + 1)$.
Steps to Solve
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Identify the coefficients For the quadratic expression $2w^2 + 11w + 9$, the coefficients are:
- $a = 2$ (coefficient of $w^2$)
- $b = 11$ (coefficient of $w$)
- $c = 9$ (constant term)
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Multiply (a) and (c) Calculate the product of the coefficients (a) and (c): $$ ac = 2 \cdot 9 = 18 $$
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Find two numbers that multiply to (ac) and add to (b) We need two numbers that multiply to $18$ and add up to $11$. The numbers are $2$ and $9$, since: $$ 2 + 9 = 11 $$ $$ 2 \cdot 9 = 18 $$
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Rewrite the middle term Replace $11w$ with $2w + 9w$: $$ 2w^2 + 2w + 9w + 9 $$
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Factor by grouping Group the terms: $$ (2w^2 + 2w) + (9w + 9) $$ Factor out common factors from each group: $$ 2w(w + 1) + 9(w + 1) $$
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Factor out the common binomial The expression can now be factored as: $$ (2w + 9)(w + 1) $$
The completely factored form of the expression is $(2w + 9)(w + 1)$.
More Information
Factoring quadratics is a key skill in algebra, allowing you to simplify expressions and solve equations. The method used here is often referred to as “factoring by grouping” after splitting the middle term based on the product and sum of coefficients.
Tips
- Incorrect pairing of factors: Sometimes students may choose numbers that do not correctly add to (b) or multiply to (ac). Double-checking these conditions is essential.
- Forgetting to factor out the common terms correctly: Ensure that all terms are taken into account when grouping and factoring.
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