To solve Part A, we need to factor the quadratic equation \( 15x^2 + 13x + 2 = 0 \).
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Start by identifying two numbers that multiply to \( 15 \times 2 = 30 \) and add up to \( 13 \). The numbers that satisfy this are \( 10 \) and \( 3 \).
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Rewrite the equation using these numbers:
\[ 15x^2 + 10x + 3x + 2 = 0 \]
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Now, group the terms:
\[ (15x^2 + 10x) + (3x + 2) = 0 \ 5x(3x + 2) + 1(3x + 2) = 0 \]
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Factor by grouping:
\[ (5x + 1)(3x + 2) = 0 \]
The correct factoring for Part A is:
(5x + 2)(3x + 1) = 0.
Now, for Part B, we find the solutions to the equation \( (5x + 2)(3x + 1) = 0 \).
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Set each factor to zero:
\[ 5x + 2 = 0 \quad \Rightarrow \quad 5x = -2 \quad \Rightarrow \quad x = -\frac{2}{5} \] \[ 3x + 1 = 0 \quad \Rightarrow \quad 3x = -1 \quad \Rightarrow \quad x = -\frac{1}{3} \]
Thus, the solutions to the equation \( 15x^2 + 13x + 2 = 0 \) are:
x = -\frac{2}{5} and x = -\frac{1}{3}.
None of the provided response options match the correct answers from the analysis. It seems likely that there might be an error either in the provided options or possibly a misunderstanding about the original equation coefficients.
If looking for roots that are in the forms presented:
Check each option:
- x = 23 and x = 15
- x = -23 and x = 15
- x = -23 and x = -15
- x = 23 and x = -15
Since neither the factored results nor the roots equate to the forms listed, please confirm the original equation or refer to the full details of the options provided.
1 answer