Multiply using the Binomial Squares Pattern.
$\left(x+\frac{3}{5}\right)^2=\square$
Answer (1)
Identify a and b in the binomial ( a + b ) : a = x and b = 5 3 .
Apply the binomial squares pattern: ( a + b ) 2 = a 2 + 2 ab + b 2 .
Substitute a = x and b = 5 3 into the pattern: ( x + 5 3 ) 2 = x 2 + 2 ( x ) ( 5 3 ) + ( 5 3 ) 2 .
Simplify the expression: x 2 + 5 6 x + 25 9 .
Explanation
Understanding the Problem We are asked to expand the square of a binomial using the binomial squares pattern. The binomial is ( x + 5 3 ) . The binomial squares pattern is ( a + b ) 2 = a 2 + 2 ab + b 2 .
Objective We want to expand the expression ( x + 5 3 ) 2 using the binomial squares pattern.
Solution Plan First, we identify a and b in the binomial ( a + b ) . In this case, a = x and b = 5 3 . Next, we apply the binomial squares pattern: ( a + b ) 2 = a 2 + 2 ab + b 2 . Then, we substitute a = x and b = 5 3 into the pattern: ( x + 5 3 ) 2 = x 2 + 2 ( x ) ( 5 3 ) + ( 5 3 ) 2 . Finally, we simplify the expression: x 2 + 5 6 x + 25 9 .
Applying the Pattern and Simplifying Let's apply the binomial squares pattern to expand the given expression. The binomial squares pattern states that ( a + b ) 2 = a 2 + 2 ab + b 2 In our case, a = x and b = 5 3 . Substituting these values into the formula, we get: ( x + 5 3 ) 2 = x 2 + 2 ⋅ x ⋅ 5 3 + ( 5 3 ) 2 Now, let's simplify each term: x 2 = x 2 2 ⋅ x ⋅ 5 3 = 5 6 x ( 5 3 ) 2 = 5 2 3 2 = 25 9 So, the expanded expression is: x 2 + 5 6 x + 25 9
Final Answer The expanded form of ( x + 5 3 ) 2 is x 2 + 5 6 x + 25 9
Examples
The binomial squares pattern is useful in many areas, such as physics and engineering. For example, when calculating the area of a square with side length x + 5 3 , you would use the binomial squares pattern to find the area: ( x + 5 3 ) 2 = x 2 + 5 6 x + 25 9 . This pattern helps simplify calculations and understand how different terms contribute to the final result. Understanding this pattern can also help in optimizing designs and predicting outcomes in various real-world scenarios.
