Factor the following expression: [tex]10 x^2+17 x+3[/tex]
Answer (1)
Find two numbers that multiply to 10 × 3 = 30 and add up to 17 , which are 15 and 2 .
Rewrite the middle term: 10 x 2 + 17 x + 3 = 10 x 2 + 15 x + 2 x + 3 .
Factor by grouping: ( 10 x 2 + 15 x ) + ( 2 x + 3 ) = 5 x ( 2 x + 3 ) + 1 ( 2 x + 3 ) .
Factor out the common binomial: ( 5 x + 1 ) ( 2 x + 3 ) . The factored form is ( 5 x + 1 ) ( 2 x + 3 ) .
Explanation
Understanding the Problem We are given the quadratic expression 10 x 2 + 17 x + 3 and asked to factor it. Factoring means expressing the quadratic as a product of two binomials, which will have the form ( a x + b ) ( c x + d ) , where a , b , c , and d are integers. Our goal is to find these integers.
Finding the Right Numbers To factor the quadratic expression 10 x 2 + 17 x + 3 , we look for two numbers that multiply to 10 × 3 = 30 and add up to 17 . These numbers are 15 and 2 , since 15 × 2 = 30 and 15 + 2 = 17 .
Rewriting the Middle Term Now we rewrite the middle term of the quadratic using these numbers: 10 x 2 + 17 x + 3 = 10 x 2 + 15 x + 2 x + 3 .
Grouping Terms Next, we factor by grouping. We group the first two terms and the last two terms: ( 10 x 2 + 15 x ) + ( 2 x + 3 ) .
Factoring Each Group We factor out the greatest common factor from each group. From the first group, 10 x 2 + 15 x , we can factor out 5 x , which gives 5 x ( 2 x + 3 ) . From the second group, 2 x + 3 , we can factor out 1 , which gives 1 ( 2 x + 3 ) . So we have 5 x ( 2 x + 3 ) + 1 ( 2 x + 3 ) .
Factoring Out the Common Binomial Now we factor out the common binomial factor, which is ( 2 x + 3 ) . This gives us ( 5 x + 1 ) ( 2 x + 3 ) .
Final Factorization Therefore, the factored form of the expression 10 x 2 + 17 x + 3 is ( 5 x + 1 ) ( 2 x + 3 ) .
Verification To verify our factorization, we expand ( 5 x + 1 ) ( 2 x + 3 ) : ( 5 x + 1 ) ( 2 x + 3 ) = 5 x ( 2 x ) + 5 x ( 3 ) + 1 ( 2 x ) + 1 ( 3 ) = 10 x 2 + 15 x + 2 x + 3 = 10 x 2 + 17 x + 3. This matches the original expression, so our factorization is correct.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden and know the area can be expressed as 10 x 2 + 17 x + 3 , factoring this expression into ( 5 x + 1 ) ( 2 x + 3 ) tells you the possible dimensions of the garden. If x represents a certain unit of length, then ( 5 x + 1 ) and ( 2 x + 3 ) would be the length and width of the garden, helping you plan its layout.
