Select the correct answer.
Which expression is equivalent to the given expression?
[tex]$2 x^2-14 x+24$[/tex]
A. [tex]$2(x-3)(x-4)$[/tex]
B. [tex]$2(x-5)(x-2)$[/tex]
C. [tex]$2(x-8)(x+3)$[/tex]
D. [tex]$(2 x-12)(x-2)$[/tex]
Answer (1)
To find the equivalent expression to 2 x 2 − 14 x + 24 , we:
Factor out the common factor 2: 2 ( x 2 − 7 x + 12 ) .
Factor the quadratic expression: 2 ( x − 3 ) ( x − 4 ) .
The equivalent expression is: 2 ( x − 3 ) ( x − 4 ) .
Explanation
Understanding the Problem We are given the quadratic expression 2 x 2 − 14 x + 24 and asked to find an equivalent expression from the given options.
Factoring out the Common Factor First, we can factor out the common factor 2 from the quadratic expression:
2 x 2 − 14 x + 24 = 2 ( x 2 − 7 x + 12 )
Factoring the Quadratic Expression Now, we need to factor the quadratic expression inside the parenthesis: x 2 − 7 x + 12 . We are looking for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4.
Finding the Factors Therefore, the factored form of the quadratic expression is ( x − 3 ) ( x − 4 ) .
The Equivalent Expression So, the equivalent expression is 2 ( x − 3 ) ( x − 4 ) .
Selecting the Correct Answer Comparing this result with the given options, we see that option A, 2 ( x − 3 ) ( x − 4 ) , matches our result.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By factoring the quadratic equation, they can determine the launch angle and initial velocity required to hit a specific target. Similarly, architects use quadratic equations to design arches and other curved structures, ensuring that they are structurally sound and aesthetically pleasing. Factoring helps in simplifying these equations to find critical parameters.
