Factor [tex]6 x^2-3 x-45[/tex]
A. [tex]3(2 x+3)(x-5)[/tex]
B. [tex]3(2 x+5)(x-3)[/tex]
C. [tex]3(2 x-5)(x+3)[/tex]
D. [tex]3(2 x-3)(x-5)[/tex]
Answer (2)
Factor out the greatest common divisor (GCD) 3: 3 ( 2 x 2 − x − 15 ) .
Find two numbers that multiply to -30 and add up to -1: -6 and 5.
Rewrite the middle term: 2 x 2 − 6 x + 5 x − 15 .
Factor by grouping: 3 ( 2 x + 5 ) ( x − 3 ) .
3 ( 2 x + 5 ) ( x − 3 )
Explanation
Initial Analysis First, we need to factor the quadratic expression 6 x 2 − 3 x − 45 .
Factoring out the GCD We can start by factoring out the greatest common divisor (GCD) of the coefficients, which is 3. This gives us 3 ( 2 x 2 − x − 15 ) .
Finding the Right Numbers Now, we need to factor the quadratic 2 x 2 − x − 15 . We are looking for two numbers that multiply to 2 × ( − 15 ) = − 30 and add up to − 1 . These numbers are − 6 and 5 .
Rewriting the Middle Term We can rewrite the middle term using these numbers: 2 x 2 − 6 x + 5 x − 15 .
Factoring by Grouping Now, we factor by grouping: 2 x ( x − 3 ) + 5 ( x − 3 ) .
Factoring out the Common Factor We factor out the common binomial factor ( x − 3 ) : ( 2 x + 5 ) ( x − 3 ) .
Complete Factorization Therefore, the complete factorization is 3 ( 2 x + 5 ) ( x − 3 ) .
Final Answer Comparing this with the given options, we see that the correct factorization is 3 ( 2 x + 5 ) ( x − 3 ) .
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. Imagine designing a bridge; you might use a quadratic equation to model the load on the bridge, and factoring would help you find the critical points where the load is highest, ensuring the bridge is safe and stable. Similarly, in business, factoring can help optimize costs and maximize profits by finding the break-even points in cost-revenue models. Factoring is also essential in physics for solving problems related to projectile motion and energy conservation.
The expression 6 x 2 − 3 x − 45 can be factored as 3 ( 2 x + 5 ) ( x − 3 ) . This involves factoring out the greatest common divisor of the coefficients first and then factoring the resulting quadratic expression. Therefore, the correct answer is option B.
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