Rewrite the expression $6 x^2+13 x+6$ by applying an appropriate structure. The expression $6 x^2+13 x+6$ can be rewritten as $\square$.

Find two numbers whose product is 6 ⋅ 6 = 36 and whose sum is 13, which are 4 and 9.
Rewrite the middle term: 6 x 2 + 13 x + 6 = 6 x 2 + 4 x + 9 x + 6 .
Factor by grouping: 6 x 2 + 4 x + 9 x + 6 = 2 x ( 3 x + 2 ) + 3 ( 3 x + 2 ) .
Factor out the common binomial factor: ( 3 x + 2 ) ( 2 x + 3 ) . The expression 6 x 2 + 13 x + 6 can be rewritten as ( 3 x + 2 ) ( 2 x + 3 ) ​ .

Explanation

Understanding the Problem We are given the quadratic expression 6 x 2 + 13 x + 6 and asked to rewrite it by applying an appropriate structure. This suggests we should factor the quadratic expression.

Finding the Right Numbers To factor the quadratic expression 6 x 2 + 13 x + 6 , we look for two numbers whose product is 6 × 6 = 36 and whose sum is 13 .

Identifying the Numbers The two numbers that satisfy these conditions are 4 and 9 , since 4 × 9 = 36 and 4 + 9 = 13 .

Rewriting the Middle Term Now we rewrite the middle term using these two numbers: 6 x 2 + 13 x + 6 = 6 x 2 + 4 x + 9 x + 6.

Factoring by Grouping Next, we factor by grouping: 6 x 2 + 4 x + 9 x + 6 = 2 x ( 3 x + 2 ) + 3 ( 3 x + 2 ) .

Factoring out the Common Factor Finally, we factor out the common binomial factor ( 3 x + 2 ) : 2 x ( 3 x + 2 ) + 3 ( 3 x + 2 ) = ( 3 x + 2 ) ( 2 x + 3 ) .

Final Answer Therefore, the expression 6 x 2 + 13 x + 6 can be rewritten as ( 3 x + 2 ) ( 2 x + 3 ) .


Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area can be expressed as 6 x 2 + 13 x + 6 square feet. By factoring this expression into ( 3 x + 2 ) ( 2 x + 3 ) , you determine that the dimensions of the garden could be ( 3 x + 2 ) feet by ( 2 x + 3 ) feet. This allows you to plan the layout of your garden based on the variable x , which could represent a design parameter or constraint.

Answered by GinnyAnswer | 2025-07-08

The expression 6 x 2 + 13 x + 6 can be factored as ( 3 x + 2 ) ( 2 x + 3 ) . This is achieved by finding two numbers that multiply to 36 and add to 13 , rewriting the expression, and factoring by grouping. Ultimately, the expression is rewritten as ( 3 x + 2 ) ( 2 x + 3 ) .
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Answered by Anonymous | 2025-08-12