Factor the quadratic $5 x^2+9 x-2$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (1)
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 + 10 x − x − 2 .
Factor by grouping: 5 x ( x + 2 ) − 1 ( x + 2 ) .
Factor out the common factor: ( 5 x − 1 ) ( x + 2 ) . The answer is ( x + 2 ) or ( 5 x − 1 ) .
Explanation
Understanding the Problem We are given the quadratic expression 5 x 2 + 9 x − 2 . Our goal is to factor this quadratic expression into two binomial factors.
Finding the Right Numbers To factor the quadratic 5 x 2 + 9 x − 2 , we look for two numbers that multiply to 5 × ( − 2 ) = − 10 and add up to 9 .
Identifying the Numbers The two numbers that satisfy these conditions are 10 and − 1 , since 10 × ( − 1 ) = − 10 and 10 + ( − 1 ) = 9 .
Rewriting the Middle Term Now we rewrite the middle term using these two numbers: 5 x 2 + 10 x − x − 2 .
Factoring by Grouping Next, we factor by grouping: 5 x ( x + 2 ) − 1 ( x + 2 ) .
Factoring Out the Common Factor We factor out the common factor ( x + 2 ) : ( 5 x − 1 ) ( x + 2 ) .
Final Answer Therefore, the factored form of the quadratic 5 x 2 + 9 x − 2 is ( 5 x − 1 ) ( x + 2 ) . We can list either of these factors as the answer.
Examples
Factoring quadratics is a fundamental skill in algebra with numerous real-world applications. For instance, engineers use factoring to design structures and calculate stress distributions. Imagine designing a bridge; factoring helps determine the optimal dimensions and materials to ensure stability and safety. Similarly, in physics, factoring can simplify equations describing projectile motion, making it easier to predict the trajectory of a ball or rocket. These applications highlight the practical importance of mastering factoring techniques.
