Simplify the following expression.
[tex]$\begin{array}{c}
(y+4)^2 \\
y^2+[?] y+
\end{array}$[/tex]
Answer (1)
Expand the expression ( y + 4 ) 2 using the binomial formula: ( y + 4 ) 2 = y 2 + 2 ( y ) ( 4 ) + 4 2 .
Calculate the coefficient of the y term: 2 × 4 = 8 .
Calculate the constant term: 4 2 = 16 .
The simplified expression is y 2 + 8 y + 16 .
Explanation
Understanding the Problem We are given an incomplete expansion of the square of a binomial expression ( y + 4 ) 2 . Our goal is to find the missing coefficient of the y term and the constant term in the expansion.
Applying the Binomial Formula To simplify the expression ( y + 4 ) 2 , we can use the binomial formula or direct multiplication. The binomial formula states that ( a + b ) 2 = a 2 + 2 ab + b 2 . In our case, a = y and b = 4 , so we have ( y + 4 ) 2 = y 2 + 2 ( y ) ( 4 ) + 4 2 .
Calculating the Missing Terms Now, we simplify the expression. First, we calculate the coefficient of the y term: 2 × 4 = 8 . Then, we calculate the constant term: 4 2 = 16 . Therefore, the complete expression is y 2 + 8 y + 16 .
Final Answer Thus, the missing coefficient of the y term is 8, and the missing constant term is 16. The simplified expression is:
( y + 4 ) 2 = y 2 + 8 y + 16
So, the complete expression is y 2 + 8 y + 16 .
Examples
Understanding how to expand binomial expressions like ( y + 4 ) 2 is useful in various real-life scenarios. For instance, if you're planning a square garden with sides of length y + 4 feet, you might want to calculate the total area. Expanding ( y + 4 ) 2 gives you y 2 + 8 y + 16 , which tells you how much area you need based on the variable y . This could help you determine the amount of soil, fencing, or other materials required for your garden. Similarly, in physics, this expansion can be used to model projectile motion or other quadratic relationships.
