Factor the quadratic $6 x^2-5 x-6$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (1)
Find two numbers that multiply to $6
\times -6 = -36 an d a dd u pt o -5 , w hi c ha re -9$ and 4 .
Rewrite the middle term: 6 x 2 − 9 x + 4 x − 6 .
Factor by grouping: 3 x ( 2 x − 3 ) + 2 ( 2 x − 3 ) .
Factor out the common binomial factor: ( 2 x − 3 ) ( 3 x + 2 ) . The answer is ( 2 x − 3 ) .
Explanation
Problem Analysis We are given the quadratic expression 6 x 2 − 5 x − 6 . Our goal is to factor this expression into two binomials.
Finding the Right Numbers We need to find two numbers that multiply to 6 × − 6 = − 36 and add up to − 5 . These numbers will help us rewrite the middle term of the quadratic.
Rewriting the Middle Term The two numbers are − 9 and 4 because − 9 × 4 = − 36 and − 9 + 4 = − 5 . Now we rewrite the middle term using these numbers: 6 x 2 − 9 x + 4 x − 6
Factoring by Grouping Next, we factor by grouping. From the first two terms, we can factor out 3 x , and from the last two terms, we can factor out 2 : 3 x ( 2 x − 3 ) + 2 ( 2 x − 3 )
Factoring out the Common Factor Now we factor out the common binomial factor ( 2 x − 3 ) : ( 2 x − 3 ) ( 3 x + 2 )
Identifying the Factors Therefore, the factored form of the quadratic is ( 2 x − 3 ) ( 3 x + 2 ) . We can list either of these factors as the answer.
Examples
Factoring quadratics is useful in many real-world scenarios, such as optimizing areas or solving projectile motion problems. For example, if you want to design a rectangular garden with a specific area and relationship between the sides, you might use factoring to find the dimensions that satisfy your requirements. Factoring helps simplify complex expressions and find solutions to practical problems involving quadratic relationships.
