Question
Factor. \[ 6 x^{2}+13 x+5 \] Select the correct choice below and, if necessary, fill in the answer box within your choice. A. \( 6 x^{2}+13 x+5=\square \). (Factor completely.) B. The trinomial is not factorable.
Ask by Pierce Fowler. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The trinomial factors completely as \( 6x^{2} + 13x + 5 = (2x + 1)(3x + 5) \).
Solution
To factor 6x² + 13x + 5, we look for two numbers that multiply to (6×5) = 30 and add up to 13. The two numbers 3 and 10 work since 3 × 10 = 30 and 3 + 10 = 13.
Now, rewrite the middle term:
6x² + 3x + 10x + 5
Group the terms:
(6x² + 3x) + (10x + 5)
Factor out the common factors in each group:
3x(2x + 1) + 5(2x + 1)
Notice (2x + 1) is a common factor:
(2x + 1)(3x + 5)
Thus, the complete factorization is:
6x² + 13x + 5 = (2x + 1)(3x + 5)
The correct choice is A: Factor completely.
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Simplify this solution Mind Expander
To factor the quadratic \( 6x^2 + 13x + 5 \), we look for two numbers that multiply to the product of the leading coefficient and the constant term (6 * 5 = 30) and add up to the middle coefficient (13). These numbers are 10 and 3. This allows us to rewrite the middle term: \[ 6x^2 + 10x + 3x + 5. \] Next, we group the terms: \[ (6x^2 + 10x) + (3x + 5). \] Factoring out common factors gives us: \[ 2x(3x + 5) + 1(3x + 5). \] Now, factoring out \( (3x + 5) \): \[ (3x + 5)(2x + 1). \] Thus, the complete factorization is: \[ 6x^2 + 13x + 5 = (3x + 5)(2x + 1). \]
