Factor the following expression.
$\begin{array}{c}
7 x^2+6 x-16 \\
(x+[?])(\square x-\square)
\end{array}$
Answer (1)
To factor the quadratic expression 7 x 2 + 6 x − 16 :
Set up the factored form as ( x + a ) ( b x + c ) .
Expand and equate coefficients to get b = 7 , c + ab = 6 , and a c = − 16 .
Solve for a using the quadratic formula, resulting in a = 2 .
Substitute a to find c = − 8 .
The factored form is ( x + 2 ) ( 7 x − 8 ) .
Explanation
Understanding the Problem We are given a quadratic expression 7 x 2 + 6 x − 16 and asked to factor it into the form ( x + a ) ( b x + c ) where a, b, and c are integers. We need to find the values of a, b, and c such that ( x + a ) ( b x + c ) = 7 x 2 + 6 x − 16 .
Setting up Equations Expand the expression ( x + a ) ( b x + c ) to get b x 2 + ( c + ab ) x + a c . Equate the coefficients of the corresponding terms in b x 2 + ( c + ab ) x + a c and 7 x 2 + 6 x − 16 . This gives us the equations b = 7 , c + ab = 6 , and a c = − 16 .
Substitution and Simplification Substitute b = 7 into the second equation to get c + 7 a = 6 , so c = 6 − 7 a . Substitute c = 6 − 7 a into the third equation to get a ( 6 − 7 a ) = − 16 , which simplifies to 6 a − 7 a 2 = − 16 , or 7 a 2 − 6 a − 16 = 0 .
Solving for a Solve the quadratic equation 7 a 2 − 6 a − 16 = 0 for a . We can use the quadratic formula a = 2 A − B ± B 2 − 4 A C , where A = 7 , B = − 6 , and C = − 16 . Calculate the discriminant: D = B 2 − 4 A C = ( − 6 ) 2 − 4 ( 7 ) ( − 16 ) = 36 + 448 = 484 . Then a = 2 ( 7 ) 6 ± 484 = 14 6 ± 22 . The two possible values for a are a = 14 6 + 22 = 14 28 = 2 and a = 14 6 − 22 = 14 − 16 = − 7 8 . Since we are looking for integer values, we take a = 2 .
Finding c and the Factorization Substitute a = 2 into c = 6 − 7 a to get c = 6 − 7 ( 2 ) = 6 − 14 = − 8 . Therefore, the factorization is ( x + 2 ) ( 7 x − 8 ) .
Final Answer Thus, the factored form of the given quadratic expression is ( x + 2 ) ( 7 x − 8 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in 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 design efficient algorithms. Factoring allows us to simplify complex expressions and solve equations, making it an essential tool in various fields.
