What is $64-9 x^2$ in factored form?
Answer (2)
Recognize the expression as a difference of squares: 64 − 9 x 2 .
Identify a and b such that a 2 = 64 and b 2 = 9 x 2 , so a = 8 and b = 3 x .
Apply the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) .
The factored form is ( 8 − 3 x ) ( 8 + 3 x ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 64 − 9 x 2 . This looks like a difference of squares, which has a special factorization pattern.
Identifying a and b The difference of squares pattern is a 2 − b 2 = ( a − b ) ( a + b ) . We need to identify what a and b are in our expression.
Rewriting the Expression We have 64 − 9 x 2 . We can rewrite 64 as 8 2 and 9 x 2 as ( 3 x ) 2 . So, we have 8 2 − ( 3 x ) 2 .
Applying the Difference of Squares Now we can see that a = 8 and b = 3 x . Plugging these into the difference of squares pattern, we get ( 8 − 3 x ) ( 8 + 3 x ) .
Final Answer Therefore, the factored form of 64 − 9 x 2 is ( 8 − 3 x ) ( 8 + 3 x ) .
Examples
Factoring the difference of squares is useful in many areas, such as simplifying algebraic expressions, solving equations, and even in engineering. For example, if you are designing a square garden with an area of 64 m 2 and want to reduce the length by 3 x meters on one side and increase the length by 3 x meters on the other side, the new area can be represented as ( 8 − 3 x ) ( 8 + 3 x ) . Factoring helps you understand how changes in dimensions affect the area.
The expression 64 − 9 x 2 can be factored using the difference of squares formula. It simplifies to ( 8 − 3 x ) ( 8 + 3 x ) . This method is useful for factorizations in algebra.
;
