What is $g^2-36$ in factored form?
Answer (2)
Recognize the expression as a difference of squares.
Apply the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) .
Identify a = g and b = 6 .
Factor the expression: g 2 − 36 = ( g − 6 ) ( g + 6 ) . The factored form is ( g − 6 ) ( g + 6 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression g 2 − 36 . This looks like a difference of squares, which has a specific factorization pattern.
Applying the Difference of Squares The difference of squares factorization formula is a 2 − b 2 = ( a − b ) ( a + b ) . We need to identify what 'a' and 'b' are in our expression.
Identifying 'a' and 'b' In our case, g 2 corresponds to a 2 , so a = g . And 36 corresponds to b 2 , so b = 36 = 6 .
Factoring the Expression Now we can substitute a = g and b = 6 into the difference of squares formula: g 2 − 36 = ( g − 6 ) ( g + 6 ) .
Final Answer Therefore, the factored form of g 2 − 36 is ( g − 6 ) ( g + 6 ) .
Examples
Factoring the difference of squares is useful in many areas, such as simplifying algebraic expressions, solving equations, and even in engineering when dealing with vibrations or oscillations. For example, if you have an equation like x 2 − 9 = 0 , you can factor it as ( x − 3 ) ( x + 3 ) = 0 , which quickly gives you the solutions x = 3 and x = − 3 . This technique makes solving certain types of equations much easier and faster.
The expression g 2 − 36 can be factored using the difference of squares formula, resulting in ( g − 6 ) ( g + 6 ) . This technique simplifies expressions and helps in solving equations. Understanding this concept is crucial for algebra and higher-level math.
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