Factor the quadratic [tex]$5 x^2-9 x-2$[/tex]. List ONLY ONE of the factors with parentheses and no spaces. Example: [tex]$(x+1)$[/tex]
Answer (2)
Find two numbers that multiply to 5 × − 2 = − 10 and add up to − 9 , which are − 10 and 1 .
Rewrite the middle term: 5 x 2 − 9 x − 2 = 5 x 2 − 10 x + x − 2 .
Factor by grouping: 5 x ( x − 2 ) + 1 ( x − 2 ) .
Factor out the common factor: ( 5 x + 1 ) ( x − 2 ) . One of the factors is ( x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 5 x 2 − 9 x − 2 . Our goal is to factor this expression into two binomials.
Finding the Right Numbers To factor the quadratic expression 5 x 2 − 9 x − 2 , we need to find two numbers that multiply to 5 × − 2 = − 10 and add up to − 9 .
Rewriting the Middle Term The two numbers that satisfy these conditions are − 10 and 1 , since ( − 10 ) × ( 1 ) = − 10 and ( − 10 ) + ( 1 ) = − 9 . Now we rewrite the middle term of the quadratic using these two numbers: 5 x 2 − 9 x − 2 = 5 x 2 − 10 x + x − 2
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 5 x 2 − 10 x ) + ( x − 2 ) Now, we factor out the greatest common factor from each group. From the first group, we can factor out 5 x , and from the second group, we can factor out 1 : 5 x ( x − 2 ) + 1 ( x − 2 )
Factoring out the Common Factor Now we see that ( x − 2 ) is a common factor in both terms, so we factor it out: ( 5 x + 1 ) ( x − 2 ) Thus, the factored form of the quadratic expression is ( 5 x + 1 ) ( x − 2 ) .
Identifying the Factors The factors of the quadratic 5 x 2 − 9 x − 2 are ( 5 x + 1 ) and ( x − 2 ) . We can list either of these factors as the answer.
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 stability. In business, factoring can be used to analyze revenue and cost functions to determine break-even points. Imagine you are designing a rectangular garden with an area represented by the quadratic expression 5 x 2 − 9 x − 2 . By factoring this expression, you can determine the possible dimensions (length and width) of the garden in terms of x . This allows you to optimize the garden's layout based on the available space and resources.
The quadratic expression 5 x 2 − 9 x − 2 can be factored as ( 5 x + 1 ) ( x − 2 ) , and one of the factors is ( 5 x + 1 ) .
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