Which will result in a perfect square trinomial?
$(3 x-5)(3 x-5)$
$(3 x-5)(5-3 x)$
$(3 x-5)(3 x+5)$
$(3 x-5)(-3 x-5)$
Answer (1)
Expand each of the given expressions.
Identify the expression that results in a perfect square trinomial.
( 3 x − 5 ) ( 3 x − 5 ) expands to 9 x 2 − 30 x + 25 , which is a perfect square trinomial.
The expression that results in a perfect square trinomial is ( 3 x − 5 ) ( 3 x − 5 ) .
Explanation
Understanding the Problem We are given four expressions and need to determine which one results in a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, i.e., ( a x + b ) 2 or ( a x − b ) 2 .
Expanding the Expressions Let's expand each of the given expressions to identify the perfect square trinomial.
Identifying the Perfect Square Trinomial
( 3 x − 5 ) ( 3 x − 5 ) = ( 3 x − 5 ) 2 = ( 3 x ) 2 − 2 ( 3 x ) ( 5 ) + ( 5 ) 2 = 9 x 2 − 30 x + 25 . This is a perfect square trinomial.
( 3 x − 5 ) ( 5 − 3 x ) = ( 3 x ) ( 5 ) + ( 3 x ) ( − 3 x ) + ( − 5 ) ( 5 ) + ( − 5 ) ( − 3 x ) = 15 x − 9 x 2 − 25 + 15 x = − 9 x 2 + 30 x − 25 . This is not a perfect square trinomial.
( 3 x − 5 ) ( 3 x + 5 ) = ( 3 x ) 2 − ( 5 ) 2 = 9 x 2 − 25 . This is a difference of squares, not a perfect square trinomial.
( 3 x − 5 ) ( − 3 x − 5 ) = ( 3 x ) ( − 3 x ) + ( 3 x ) ( − 5 ) + ( − 5 ) ( − 3 x ) + ( − 5 ) ( − 5 ) = − 9 x 2 − 15 x + 15 x + 25 = − 9 x 2 + 25 . This is not a perfect square trinomial.
Conclusion Therefore, the first expression, ( 3 x − 5 ) ( 3 x − 5 ) , results in a perfect square trinomial: 9 x 2 − 30 x + 25 .
Examples
Perfect square trinomials are useful in various applications, such as completing the square to solve quadratic equations, simplifying algebraic expressions, and modeling physical phenomena. For example, in physics, the equation of motion for a simple harmonic oscillator can sometimes be expressed using a perfect square trinomial, making it easier to analyze the system's behavior. In engineering, perfect square trinomials can arise in the design of filters and control systems, where they help ensure stability and desired performance characteristics. Understanding and recognizing perfect square trinomials can greatly simplify these types of problems.
