Factor the following expression:
[tex]\begin{array}{c}
11 x^2+74 x+48 \\
(11 x+[?])(x \square 6)
\end{array}[/tex]
Answer (2)
We are given the quadratic expression 11 x 2 + 74 x + 48 and the partially factored form ( 11 x + [ ?]) ( x □ 6 ) .
We assume the factored form is ( 11 x + a ) ( x + b ) and expand it to get 11 x 2 + ( 11 b + a ) x + ab .
Comparing coefficients, we have 11 b + a = 74 and ab = 48 . We consider b = 6 .
Substituting b = 6 into 11 b + a = 74 , we get a = 8 . Thus, the factored form is ( 11 x + 8 ) ( x + 6 ) .
The final answer is 8 .
Explanation
Understanding the Problem We are given a quadratic expression 11 x 2 + 74 x + 48 to factor. We are also given a partially factored form ( 11 x + [ ?]) ( x □ 6 ) , where we need to find the missing constant term and the sign □ .
Expanding the Factored Form Let's assume the factored form is ( 11 x + a ) ( x + b ) , where a and b are constants to be determined. Expanding this expression, we get 11 x 2 + 11 b x + a x + ab = 11 x 2 + ( 11 b + a ) x + ab .
Comparing Coefficients Comparing the coefficients of the expanded form with the given quadratic expression 11 x 2 + 74 x + 48 , we have two equations:
11 b + a = 74 and ab = 48 .
Considering b = 6 Since one of the factors is given as ( x □ 6 ) , we can assume b = 6 or b = − 6 . Let's consider the case where b = 6 .
Solving for a Substituting b = 6 into the equation 11 b + a = 74 , we get 11 ( 6 ) + a = 74 , which simplifies to 66 + a = 74 . Solving for a , we find a = 74 − 66 = 8 .
Checking the Solution Now, let's check if ab = 48 holds true with a = 8 and b = 6 . We have 8 ( 6 ) = 48 , which is true. Thus, the factored form is ( 11 x + 8 ) ( x + 6 ) .
Considering b = -6 If we consider the case where b = − 6 , then 11 ( − 6 ) + a = 74 , so − 66 + a = 74 , which means a = 140 . Then ab = ( 140 ) ( − 6 ) = − 840 , which is not equal to 48. So, this case is invalid.
Final Answer Therefore, the factored form is ( 11 x + 8 ) ( x + 6 ) . The missing constant term is 8, and the sign is +.
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. Factoring helps simplify complex equations, making them easier to solve and understand. In construction, if you need to build a rectangular structure with a specific area, factoring can help you determine the possible dimensions of the structure. For instance, if you want the area to be represented by the quadratic 11 x 2 + 74 x + 48 , factoring it into ( 11 x + 8 ) ( x + 6 ) gives you possible expressions for the length and width of the structure.
The expression 11 x 2 + 74 x + 48 factors to ( 11 x + 8 ) ( x + 6 ) . The missing value is 8 and the sign is positive. Thus, the final answer is determined to be 8 .
;
