What is $64-9 x^2$ in factored form?
Answer (2)
Recognize the expression as a difference of squares.
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: 64 − 9 x 2 = ( 8 − 3 x ) ( 8 + 3 x ) .
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 . This means 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 ) .
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 to analyze vibrations or oscillations. For example, if you have a structure oscillating with a displacement described by 64 − 9 t 2 , where t is time, factoring it into ( 8 − 3 t ) ( 8 + 3 t ) helps identify the times when the displacement is zero, indicating key points in the oscillation.
The expression 64 − 9 x 2 can be factored as ( 8 − 3 x ) ( 8 + 3 x ) by recognizing it as a difference of squares. This involves identifying 64 as 8 2 and 9 x 2 as ( 3 x ) 2 , then applying the difference of squares formula. Thus, the final factored form is ( 8 − 3 x ) ( 8 + 3 x ) .
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