What is the completely factored form of [tex]$6 x^2-13 x-5$[/tex]?
A. [tex]$(2 x-5)(3 x+1)$[/tex]
B. [tex]$(2 x+5)(3 x-1)$[/tex]
C. [tex]$(2 x-1)(3 x-5)$[/tex]
D. [tex]$(2 x+1)(3 x+5)$[/tex]
Answer (1)
Find two numbers that multiply to 6 × − 5 = − 30 and add up to − 13 , which are − 15 and 2 .
Rewrite the middle term: 6 x 2 − 13 x − 5 = 6 x 2 − 15 x + 2 x − 5 .
Factor by grouping: 6 x 2 − 15 x + 2 x − 5 = 3 x ( 2 x − 5 ) + 1 ( 2 x − 5 ) = ( 2 x − 5 ) ( 3 x + 1 ) .
The completely factored form is ( 2 x − 5 ) ( 3 x + 1 ) .
Explanation
Understanding the Problem We are given the quadratic expression 6 x 2 − 13 x − 5 and asked to factor it completely.
Finding the Right Numbers To factor this quadratic, we look for two numbers that multiply to 6 × − 5 = − 30 and add up to − 13 .
Identifying the Factors The two numbers are − 15 and 2 , since ( − 15 ) ( 2 ) = − 30 and − 15 + 2 = − 13 .
Rewriting the Middle Term Now we rewrite the middle term of the quadratic using these two numbers: 6 x 2 − 13 x − 5 = 6 x 2 − 15 x + 2 x − 5
Factoring by Grouping Next, we factor by grouping: 6 x 2 − 15 x + 2 x − 5 = 3 x ( 2 x − 5 ) + 1 ( 2 x − 5 )
Factoring out the Common Factor Now we factor out the common binomial factor ( 2 x − 5 ) : 3 x ( 2 x − 5 ) + 1 ( 2 x − 5 ) = ( 2 x − 5 ) ( 3 x + 1 )
Checking the Answer Finally, we check our factored form by expanding it: ( 2 x − 5 ) ( 3 x + 1 ) = 2 x ( 3 x ) + 2 x ( 1 ) − 5 ( 3 x ) − 5 ( 1 ) = 6 x 2 + 2 x − 15 x − 5 = 6 x 2 − 13 x − 5 This matches the original quadratic, so our factored form is correct.
Final Answer Therefore, the completely factored form of 6 x 2 − 13 x − 5 is ( 2 x − 5 ) ( 3 x + 1 ) .
Examples
Factoring quadratic expressions is useful in many real-world applications, such as optimizing areas or volumes. For example, suppose you want to design a rectangular garden with an area represented by the expression 6 x 2 − 13 x − 5 . By factoring this expression into ( 2 x − 5 ) ( 3 x + 1 ) , you can determine the possible dimensions of the garden in terms of x . This allows you to explore different garden layouts and choose the one that best fits your needs and space.
