Select the expression that is equivalent to $(x+5)^2$.
A. $x^2+10 x+25$
B. $x^2+10 x^2+25$
C. $x^2+5 x+25$
D. $2 x^2+10 x+25$
Answer (1)
Expand the expression ( x + 5 ) 2 using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 .
Substitute a = x and b = 5 into the formula: ( x + 5 ) 2 = x 2 + 2 ( x ) ( 5 ) + 5 2 .
Simplify the expression: x 2 + 10 x + 25 .
The equivalent expression is x 2 + 10 x + 25 .
Explanation
Understanding the problem We are asked to find the expression that is equivalent to ( x + 5 ) 2 . This means we need to expand the expression and simplify it.
Applying the binomial formula To expand ( x + 5 ) 2 , we can use the formula ( a + b ) 2 = a 2 + 2 ab + b 2 . In this case, a = x and b = 5 .
Substitution Substituting a = x and b = 5 into the formula, we get: ( x + 5 ) 2 = x 2 + 2 ( x ) ( 5 ) + 5 2
Simplifying the expression Now, we simplify the expression: x 2 + 2 ( x ) ( 5 ) + 5 2 = x 2 + 10 x + 25
Finding the matching option Comparing the expanded expression x 2 + 10 x + 25 with the given options, we see that it matches option A.
Final Answer Therefore, the expression equivalent to ( x + 5 ) 2 is x 2 + 10 x + 25 .
Examples
Understanding how to expand squared binomials like ( x + 5 ) 2 is useful in many areas, such as physics and engineering. For example, when calculating the area of a square with side length x + 5 , you would use this expansion. If x represents a length of 2 meters, then the side length is 7 meters, and the area is ( 2 + 5 ) 2 = 7 2 = 49 square meters. Expanding the expression, we get x 2 + 10 x + 25 = 2 2 + 10 ( 2 ) + 25 = 4 + 20 + 25 = 49 square meters, which confirms the result.
