Multiply using the Binomial Squares Pattern

$(a-5)^2=$

Recognize the binomial squares pattern: ( a − b ) 2 = a 2 − 2 ab + b 2 .
Apply the pattern with a = a and b = 5 : ( a − 5 ) 2 = a 2 − 2 ⋅ a ⋅ 5 + 5 2 .
Simplify the expression: a 2 − 10 a + 25 .
The final expanded form is: a 2 − 10 a + 25 ​ .

Explanation

Understanding the Problem We need to expand ( a − 5 ) 2 using the binomial squares pattern. The binomial squares pattern is ( x − y ) 2 = x 2 − 2 x y + y 2 .

Applying the Binomial Squares Pattern Apply the binomial squares pattern ( x − y ) 2 = x 2 − 2 x y + y 2 with x = a and y = 5 . Substitute a for x and 5 for y in the formula: ( a − 5 ) 2 = a 2 − 2 ( a ) ( 5 ) + 5 2 .

Simplifying the Expression Simplify the expression:


a 2 − 2 ( a ) ( 5 ) + 5 2 = a 2 − 10 a + 25
So, ( a − 5 ) 2 = a 2 − 10 a + 25 .
Examples
The binomial squares pattern is useful in many areas, such as physics and engineering, where squared differences often appear in formulas. For example, in calculating the kinetic energy of an object or in determining the distance between two points in space, you might encounter expressions that can be simplified using this pattern. Understanding and applying the binomial squares pattern can simplify these calculations and make problem-solving more efficient.

Answered by GinnyAnswer | 2025-07-07