Factorize the expression. $9 x^2+11 x+2=0$
Answer (1)
Find two numbers whose product is 9 × 2 = 18 and whose sum is 11 , which are 9 and 2 .
Rewrite the middle term: 9 x 2 + 9 x + 2 x + 2 = 0 .
Factor by grouping: 9 x ( x + 1 ) + 2 ( x + 1 ) = 0 .
Factor out the common factor: ( 9 x + 2 ) ( x + 1 ) = 0 . The factorization is ( 9 x + 2 ) ( x + 1 ) .
Explanation
Understanding the Problem We are given the quadratic expression 9 x 2 + 11 x + 2 = 0 and asked to factorize it. Factoring involves expressing the quadratic as a product of two binomials.
Finding the Right Numbers To factorize the quadratic expression 9 x 2 + 11 x + 2 , we look for two numbers whose product equals the product of the leading coefficient (9) and the constant term (2), which is 9 × 2 = 18 , and whose sum equals the middle coefficient (11).
Splitting the Middle Term The two numbers that satisfy these conditions are 9 and 2, since 9 × 2 = 18 and 9 + 2 = 11 . We will use these numbers to split the middle term of the quadratic expression.
Rewriting the Expression Now, we rewrite the middle term using these two numbers: 9 x 2 + 9 x + 2 x + 2 = 0 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 9 x 2 + 9 x ) + ( 2 x + 2 ) = 0 .
Factoring Out Common Factors We factor out the greatest common factor from each group: 9 x ( x + 1 ) + 2 ( x + 1 ) = 0 .
Final Factorization Now, we factor out the common binomial factor ( x + 1 ) from the entire expression: ( 9 x + 2 ) ( x + 1 ) = 0 .
The Solution Therefore, the factorization of the quadratic expression 9 x 2 + 11 x + 2 is ( 9 x + 2 ) ( x + 1 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra with applications in various fields. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. Imagine you are designing a bridge and need to ensure that the forces acting on it are balanced. By expressing these forces as a quadratic equation and factoring it, you can determine the critical points where the bridge is most vulnerable, allowing you to reinforce those areas and ensure the bridge's safety. This process ensures structural integrity and prevents potential disasters.
