Question
Factor. \( 2 x^{2}-x-15 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( 2 x^{2}-x-15=\square \) (Factor completely.) B. The trinomial is not factorable.
Ask by Cruz Thornton. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 2x^{2} - x - 15 = (2x + 5)(x - 3) \)
Solution
To factor 2x² - x - 15, look for two numbers that multiply to (2 * -15) = -30 and add up to -1 (the coefficient of x). The numbers -6 and 5 work because (-6) + 5 = -1 and (-6) * 5 = -30.
Now, rewrite the middle term using these numbers:
2x² - 6x + 5x - 15
Group the terms:
(2x² - 6x) + (5x - 15)
Factor each group:
2x(x - 3) + 5(x - 3)
Now, factor out the common factor (x - 3):
(2x + 5)(x - 3)
So the factorization is:
2x² - x - 15 = (2x + 5)(x - 3)
Select Choice A with the answer: (2x + 5)(x - 3).
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Simplify this solution Extra Insights
To factor the expression \( 2x^2 - x - 15 \), we look for two numbers that multiply to \( 2 \times -15 = -30 \) and add to \( -1 \) (the coefficient of \( x \)). These two numbers are \( 5 \) and \( -6 \). Rewriting the middle term, we have: \( 2x^2 + 5x - 6x - 15 \). Now, we can factor by grouping: \( x(2x + 5) - 3(2x + 5) = (2x + 5)(x - 3) \). So, \( 2x^2 - x - 15 = (2x + 5)(x - 3) \). Now we can check our work by expanding the factors: \( (2x + 5)(x - 3) = 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15 \), confirming that it factors correctly! Therefore, the completed factorization is \( (2x + 5)(x - 3) \).
