Factor the quadratic $10 x^2+19 x-15$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (2)
Find two numbers that multiply to 10 × − 15 = − 150 and add up to 19 , which are − 6 and 25 .
Rewrite the middle term: 10 x 2 + 19 x − 15 = 10 x 2 + 25 x − 6 x − 15 .
Factor by grouping: 5 x ( 2 x + 5 ) − 3 ( 2 x + 5 ) .
Factor out the common factor: ( 5 x − 3 ) ( 2 x + 5 ) . One of the factors is ( 2 x + 5 ) .
Explanation
Problem Analysis We are given the quadratic expression 10 x 2 + 19 x − 15 . Our goal is to factor this quadratic expression into two binomials.
Finding the Right Numbers We need to find two numbers that multiply to 10 × − 15 = − 150 and add up to 19 . From the tool, we found that the two numbers are − 6 and 25 .
Rewriting the Middle Term Now, we rewrite the middle term using these two numbers: 10 x 2 + 19 x − 15 = 10 x 2 + 25 x − 6 x − 15
Factoring by Grouping Next, we factor by grouping: 10 x 2 + 25 x − 6 x − 15 = 5 x ( 2 x + 5 ) − 3 ( 2 x + 5 )
Factoring out the Common Factor Now, we factor out the common factor ( 2 x + 5 ) : 5 x ( 2 x + 5 ) − 3 ( 2 x + 5 ) = ( 5 x − 3 ) ( 2 x + 5 )
Final Answer Therefore, the factored form of the quadratic is ( 5 x − 3 ) ( 2 x + 5 ) . We are asked to list only one of the factors.
Examples
Factoring quadratics is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Computer scientists use factoring in cryptography to secure data transmissions. Financial analysts use factoring to model investment portfolios and assess risk. By mastering factoring, you'll be equipped to solve a wide range of problems in various fields.
To factor the quadratic 10 x 2 + 19 x − 15 , we rewrite it using two numbers that multiply to − 150 and add to 19 . After factoring, we find that one of the factors is ( 2 x + 5 ) .
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