Simplify the following expression.
[tex]$\begin{array}{c}
(x-4)^2 \
x^2-8 x+[?]
\end{array}$[/tex]
Answer (2)
Expand the expression ( x − 4 ) 2 using the formula ( a − b ) 2 = a 2 − 2 ab + b 2 .
Substitute a = x and b = 4 into the formula: ( x − 4 ) 2 = x 2 − 2 ( x ) ( 4 ) + 4 2 .
Simplify the expression: ( x − 4 ) 2 = x 2 − 8 x + 16 .
Identify the missing term by comparing the expanded expression to the given expression, which gives the final answer: 16 .
Explanation
Problem Analysis We are given an incomplete expansion of ( x − 4 ) 2 and asked to find the missing constant term.
Expansion Strategy To find the missing term, we need to expand the expression ( x − 4 ) 2 . We can use the formula for the square of a binomial: ( a − b ) 2 = a 2 − 2 ab + b 2 . In our case, a = x and b = 4 .
Applying the Formula Applying the formula, we get: ( x − 4 ) 2 = x 2 − 2 ( x ) ( 4 ) + 4 2
Simplifying the Expression Simplifying the expression, we have: ( x − 4 ) 2 = x 2 − 8 x + 16
Identifying the Missing Term Comparing this to the given expression x 2 − 8 x + [ ?] , we see that the missing term is 16.
Final Answer Therefore, the simplified expression is x 2 − 8 x + 16 .
Examples
Understanding how to expand binomial expressions like ( x − 4 ) 2 is useful in many areas, such as physics and engineering, where you might need to model the trajectory of a projectile. For example, if you're analyzing the height of a ball thrown in the air, the equation might involve squared terms that need to be expanded to understand the ball's motion fully. Expanding the expression allows us to easily identify key parameters, such as the maximum height or the time it takes to reach the ground. This skill is also crucial in calculus when dealing with derivatives and integrals of polynomial functions.
To simplify the expression ( x − 4 ) 2 , we expand it to find the missing term in x 2 − 8 x + [ ?] . The expansion yields x 2 − 8 x + 16 , so the missing term is 16. Therefore, the complete expression is x 2 − 8 x + 16 .
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