Factor the trinomial completely: [tex]$2 x^2+38 x-40$[/tex]
Answer (2)
Factor out the greatest common factor (GCF) of the coefficients, which is 2: 2 ( x 2 + 19 x − 20 ) .
Find two numbers that multiply to -20 and add to 19. These numbers are 20 and -1.
Factor the quadratic expression: ( x + 20 ) ( x − 1 ) .
Combine the GCF and the factored quadratic expression: 2 ( x + 20 ) ( x − 1 ) .
Explanation
Understanding the Problem We are asked to factor the trinomial 2 x 2 + 38 x − 40 completely. This means we want to express it as a product of simpler expressions.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the coefficients. The GCF of 2, 38, and -40 is 2. We factor out the 2 from the trinomial: 2 x 2 + 38 x − 40 = 2 ( x 2 + 19 x − 20 )
Factoring the Quadratic Expression Now we need to factor the quadratic expression x 2 + 19 x − 20 . We are looking for two numbers that multiply to -20 and add to 19. These numbers are 20 and -1. So, we can write the quadratic expression as: x 2 + 19 x − 20 = ( x + 20 ) ( x − 1 )
Complete Factorization Finally, we combine the GCF we factored out earlier with the factored quadratic expression to get the complete factorization of the trinomial: 2 x 2 + 38 x − 40 = 2 ( x + 20 ) ( x − 1 )
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures, ensuring stability and efficiency. In business, factoring can help optimize costs and maximize profits by breaking down complex financial models into manageable components. Even in computer science, factoring is used in cryptography to secure data and protect sensitive information.
To factor the trinomial 2 x 2 + 38 x − 40 , first factor out the GCF, which is 2, resulting in 2 ( x 2 + 19 x − 20 ) . Then, factor the quadratic part to get ( x + 20 ) ( x − 1 ) and combine it with the GCF. The final factorization is 2 ( x + 20 ) ( x − 1 ) .
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