Which expression is equivalent to $n^2+26 n+88$ for all values of $n$?
A. $(n+8)(n+11)$
B. $(n+4)(n+22)$
C. $(n+4)(n+24)$
D. $(n+8)(n+18)$
Answer (2)
The expression equivalent to n 2 + 26 n + 88 is ( n + 4 ) ( n + 22 ) , which corresponds to Option B. This was determined by expanding each option and comparing their forms to the original expression. Option B is the only one that matches perfectly.
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Expand each of the given options.
Compare the expanded form of each option with the given quadratic expression n 2 + 26 n + 88 .
Identify the option whose expanded form matches the given quadratic expression.
The expression equivalent to n 2 + 26 n + 88 is ( n + 4 ) ( n + 22 ) .
Explanation
Understanding the Problem We are given the quadratic expression n 2 + 26 n + 88 and asked to find an equivalent expression from the given options. To do this, we will expand each option and compare it to the given expression.
Expanding the Options Let's expand each of the given options:
Option 1: ( n + 8 ) ( n + 11 ) = n 2 + 11 n + 8 n + 88 = n 2 + 19 n + 88 Option 2: ( n + 4 ) ( n + 22 ) = n 2 + 22 n + 4 n + 88 = n 2 + 26 n + 88 Option 3: ( n + 4 ) ( n + 24 ) = n 2 + 24 n + 4 n + 96 = n 2 + 28 n + 96 Option 4: ( n + 8 ) ( n + 18 ) = n 2 + 18 n + 8 n + 144 = n 2 + 26 n + 144
Finding the Matching Expression Comparing the expanded forms with the given expression n 2 + 26 n + 88 , we see that Option 2, ( n + 4 ) ( n + 22 ) , matches exactly.
Conclusion Therefore, the expression equivalent to n 2 + 26 n + 88 is ( n + 4 ) ( n + 22 ) .
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 needed to hit a specific target. Similarly, architects use quadratic equations to design arches and bridges, ensuring structural stability and optimal use of materials. Understanding how to factor quadratic expressions allows them to solve for critical dimensions and load-bearing capacities.
