Factor completely: 2w² + 11w + 9.
Understand the Problem
The question is asking to completely factor the quadratic expression given by 2w² + 11w + 9. This involves finding factors of the expression that can be multiplied to recreate the original expression.
Answer
The complete factorization is \( (w + 1)(2w + 9) \).
Answer for screen readers
The completely factored form of the quadratic expression ( 2w^2 + 11w + 9 ) is
$$ (w + 1)(2w + 9) $$
Steps to Solve
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Identify the quadratic expression Recognize the given expression: ( 2w^2 + 11w + 9 ).
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Multiply the leading coefficient and the constant Multiply the coefficient of ( w^2 ) (which is 2) by the constant term (which is 9):
$$ 2 \times 9 = 18 $$
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Find two numbers that multiply to 18 and add up to 11 We need two numbers that multiply to 18 and add up to 11. The numbers that work are 2 and 9.
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Rewrite the middle term with the two numbers Rewrite the expression using the two numbers:
$$ 2w^2 + 2w + 9w + 9 $$
- Group the terms Group the terms in pairs:
$$ (2w^2 + 2w) + (9w + 9) $$
- Factor out the greatest common factor in each group From the first group ((2w^2 + 2w)), factor out (2w). From the second group ((9w + 9)), factor out (9):
$$ 2w(w + 1) + 9(w + 1) $$
- Factor out the common binomial Now, notice that ((w + 1)) is common in both terms. Factor it out:
$$ (w + 1)(2w + 9) $$
The completely factored form of the quadratic expression ( 2w^2 + 11w + 9 ) is
$$ (w + 1)(2w + 9) $$
More Information
Factoring quadratics helps in solving equations or simplifying expressions. The above methods are fundamental techniques used in algebra, crucial for various applications in mathematics and science.
Tips
- Forgetting to multiply the leading coefficient with the constant.
- Failing to break down the middle term correctly.
- Not factoring out the Common Factor properly.
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