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.
Finding the Square Roots 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 . Thus, a = 8 and b = 3 x .
Applying the Factorization Now we can apply the difference of squares factorization: 64 − 9 x 2 = ( 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, imagine you are designing a square garden with an area of 64 m 2 but need to remove a smaller square section for a patio with an area of 9 x 2 m 2 . Factoring the difference of the areas, 64 − 9 x 2 , helps you determine the dimensions of the remaining garden space, which can be represented as ( 8 − 3 x ) ( 8 + 3 x ) . This allows you to plan the layout efficiently.
The expression 64 − 9 x 2 can be factored using the difference of squares formula. It can be rewritten as ( 8 − 3 x ) ( 8 + 3 x ) . Thus, the factored form is ( 8 − 3 x ) ( 8 + 3 x ) .
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