Factor the following expression:
[tex]2 x^2-9 x+10[/tex]
Enter the correct symbol, + or -.
[tex](2 x[?] 5)(x-[?])[/tex]
Answer (2)
Test different sign combinations in the factored form.
Expand each combination to check if it matches the original quadratic expression.
Identify the correct combination: ( 2 x − 5 ) ( x − 2 ) .
The missing signs are both negative: − .
Explanation
Understanding the Problem We are given the quadratic expression 2 x 2 − 9 x + 10 and asked to factor it into the form ( 2 x ? 5 ) ( x ? 2 ) , where the question marks represent either + or − . Our goal is to determine the correct signs.
Testing Possible Factorizations Let's consider the possible combinations of signs and expand the resulting expressions to see which one matches the given quadratic.
Expanding the Factorizations
( 2 x + 5 ) ( x − 2 ) = 2 x 2 − 4 x + 5 x − 10 = 2 x 2 + x − 10 . This does not match the given expression.
( 2 x − 5 ) ( x − 2 ) = 2 x 2 − 4 x − 5 x + 10 = 2 x 2 − 9 x + 10 . This matches the given expression.
( 2 x + 5 ) ( x + 2 ) = 2 x 2 + 4 x + 5 x + 10 = 2 x 2 + 9 x + 10 . This does not match the given expression.
( 2 x − 5 ) ( x + 2 ) = 2 x 2 + 4 x − 5 x − 10 = 2 x 2 − x − 10 . This does not match the given expression.
Determining the Correct Signs From the expansions above, we see that ( 2 x − 5 ) ( x − 2 ) is the correct factorization of 2 x 2 − 9 x + 10 . Therefore, the missing signs are both negative.
Final Answer The correct factorization is ( 2 x − 5 ) ( x − 2 ) . Thus, the missing signs are both negative.
Examples
Factoring quadratic expressions 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. Similarly, economists use factoring to model economic trends and make predictions about future growth. Factoring also plays a crucial role in computer science, where it is used to optimize algorithms and solve complex problems. Understanding how to factor quadratic expressions can help you solve a wide range of problems in various fields.
The quadratic expression 2 x 2 − 9 x + 10 factors to ( 2 x − 5 ) ( x − 2 ) . This means the missing signs in the expression are both negative. The factorization matches the original expression upon expansion.
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