Factor the following expression
$\begin{array}{r}
2 x^2+19 x+17 \\
(x+[?])(\square x+\square)
\end{array}$
Answer (1)
We are asked to factor the quadratic expression 2 x 2 + 19 x + 17 .
We express the quadratic as a product of two binomials: ( x + a ) ( b x + c ) .
We expand the product and equate coefficients to obtain the equations b = 2 , c + ab = 19 , and a c = 17 .
Since 17 is prime, we test possible integer values for a and c .
We find that a = 1 , b = 2 , and c = 17 satisfy the equations, so the factored expression is ( x + 1 ) ( 2 x + 17 ) .
Explanation
Understanding the Problem We are given a quadratic expression 2 x 2 + 19 x + 17 and we want to factor it into the form ( x + a ) ( b x + c ) where a , b , c are integers. Our goal is to find the values of a , b , c such that ( x + a ) ( b x + c ) = 2 x 2 + 19 x + 17 .
Setting up Equations Expanding the expression ( x + a ) ( b x + c ) gives us b x 2 + c x + ab x + a c = b x 2 + ( c + ab ) x + a c . We need to equate the coefficients of the quadratic expression 2 x 2 + 19 x + 17 with the expanded form b x 2 + ( c + ab ) x + a c . This gives us the following equations:
b = 2
c + ab = 19
a c = 17
Finding Possible Values Since a c = 17 and 17 is a prime number, the possible integer values for a and c are ( a , c ) = ( 1 , 17 ) or ( 17 , 1 ) or ( − 1 , − 17 ) or ( − 17 , − 1 ) . We also know that b = 2 , so we can substitute this into the equation c + ab = 19 to get c + 2 a = 19 , which can be rewritten as c = 19 − 2 a .
Testing the Values Now, let's test the possible values of a and c to see which pair satisfies both a c = 17 and c = 19 − 2 a .
If a = 1 , then c = 19 − 2 ( 1 ) = 19 − 2 = 17 . This satisfies a c = ( 1 ) ( 17 ) = 17 . So, a = 1 and c = 17 is a valid solution.
If a = 17 , then c = 19 − 2 ( 17 ) = 19 − 34 = − 15 . This does not satisfy a c = ( 17 ) ( − 15 ) = − 255 e q 17 .
If a = − 1 , then c = 19 − 2 ( − 1 ) = 19 + 2 = 21 . This does not satisfy a c = ( − 1 ) ( 21 ) = − 21 e q 17 .
If a = − 17 , then c = 19 − 2 ( − 17 ) = 19 + 34 = 53 . This does not satisfy a c = ( − 17 ) ( 53 ) = − 901 e q 17 .
Final Factorization Therefore, the only valid solution is a = 1 , b = 2 , and c = 17 . The factored expression is ( x + 1 ) ( 2 x + 17 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. 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. Understanding how to factor quadratic expressions allows you to solve a wide range of problems in various fields.
