Factor.
$5 x^2-13 x+6$
Answer (1)
Find two numbers that multiply to 5 ⋅ 6 = 30 and add up to − 13 , which are − 3 and − 10 .
Rewrite the middle term: 5 x 2 − 13 x + 6 = 5 x 2 − 10 x − 3 x + 6 .
Factor by grouping: 5 x ( x − 2 ) − 3 ( x − 2 ) .
Factor out the common factor: ( 5 x − 3 ) ( x − 2 ) . The factored form is ( 5 x − 3 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 5 x 2 − 13 x + 6 and we want to factor it. Factoring means rewriting the expression as a product of simpler expressions. In this case, we want to find two binomials that multiply together to give us the original quadratic.
Finding the Right Numbers To factor the quadratic expression 5 x 2 − 13 x + 6 , we look for two numbers that multiply to 5 × 6 = 30 and add up to − 13 . These two numbers are − 3 and − 10 because ( − 3 ) × ( − 10 ) = 30 and ( − 3 ) + ( − 10 ) = − 13 .
Rewriting the Middle Term Now we rewrite the middle term of the quadratic using these two numbers: 5 x 2 − 13 x + 6 = 5 x 2 − 10 x − 3 x + 6 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 5 x 2 − 10 x ) + ( − 3 x + 6 ) . Now, we factor out the greatest common factor from each group. From the first group, we can factor out 5 x , and from the second group, we can factor out − 3 : 5 x ( x − 2 ) − 3 ( x − 2 ) .
Factoring out the Common Factor Notice that ( x − 2 ) is a common factor in both terms. We factor out ( x − 2 ) to get ( 5 x − 3 ) ( x − 2 ) . Therefore, the factored form of the quadratic expression is ( 5 x − 3 ) ( x − 2 ) .
Final Answer The factored form of the quadratic expression 5 x 2 − 13 x + 6 is ( 5 x − 3 ) ( x − 2 ) .
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 represented by the expression 5 x 2 − 13 x + 6 , where x is a variable related to the dimensions. By factoring this expression into ( 5 x − 3 ) ( x − 2 ) , you determine the possible dimensions of the garden in terms of x . This allows you to plan the layout and choose appropriate values for x to meet specific size requirements or optimize the use of space.
