Factor $12 x^2+33 x+18$.
Answer (1)
Factor out the greatest common divisor (GCD) from the quadratic expression: 12 x 2 + 33 x + 18 = 3 ( 4 x 2 + 11 x + 6 ) .
Find two numbers that multiply to 4 × 6 = 24 and add up to 11; these numbers are 3 and 8.
Rewrite the middle term and factor by grouping: 4 x 2 + 3 x + 8 x + 6 = ( 4 x + 3 ) ( x + 2 ) .
Include the common factor 3 to get the final factored form: 3 ( 4 x + 3 ) ( x + 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 12 x 2 + 33 x + 18 and asked to determine if it is factorable. If it is, we need to find its factors.
Factoring out the Common Factor First, we look for a common factor among all the terms. The greatest common divisor (GCD) of 12, 33, and 18 is 3. We can factor out 3 from the expression: 12 x 2 + 33 x + 18 = 3 ( 4 x 2 + 11 x + 6 ) Now we need to determine if the quadratic 4 x 2 + 11 x + 6 is factorable.
Finding the Right Numbers To factor the quadratic 4 x 2 + 11 x + 6 , we look for two numbers that multiply to 4 × 6 = 24 and add up to 11. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. We can see that 3 and 8 satisfy the conditions since 3 × 8 = 24 and 3 + 8 = 11 .
Factoring by Grouping Now we rewrite the middle term using these two numbers: 4 x 2 + 11 x + 6 = 4 x 2 + 3 x + 8 x + 6 Next, we factor by grouping: 4 x 2 + 3 x + 8 x + 6 = ( 4 x 2 + 8 x ) + ( 3 x + 6 ) = 4 x ( x + 2 ) + 3 ( x + 2 ) We factor out the common binomial factor ( x + 2 ) : 4 x ( x + 2 ) + 3 ( x + 2 ) = ( 4 x + 3 ) ( x + 2 ) Therefore, the factored form of 4 x 2 + 11 x + 6 is ( 4 x + 3 ) ( x + 2 ) .
Final Factored Form Finally, we include the common factor 3 that we factored out at the beginning: 3 ( 4 x + 3 ) ( x + 2 ) So, the factored form of 12 x 2 + 33 x + 18 is 3 ( 4 x + 3 ) ( x + 2 ) .
Conclusion The quadratic expression 12 x 2 + 33 x + 18 is factorable, and its factored form is 3 ( 4 x + 3 ) ( x + 2 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra with numerous real-world applications. For example, engineers use factoring to design structures and calculate stress distributions. Architects use factoring to optimize space and materials in building designs. Financial analysts use factoring to model and predict market trends. By understanding how to factor quadratic expressions, you can solve a wide range of problems in various fields.
