What is $g^2-36$ in factored form?
Answer (2)
Recognize the expression as a difference of squares.
Apply the difference of squares formula: a 2 − b 2 = ( a − b ) ( a + b ) .
Identify a = g and b = 6 .
Factor the expression: ( 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.
Stating the Formula 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 is like a 2 , so a = g . And 36 is like b 2 , so b = 36 = 6 .
Applying the Formula Now we can apply the formula: g 2 − 36 = ( g − 6 ) ( g + 6 ) .
Final Answer Therefore, the factored form of g 2 − 36 is ( g − 6 ) ( g + 6 ) .
Examples
The difference of squares factorization 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 x 2 − 9 , you can factor it as ( x − 3 ) ( x + 3 ) to find the points where the displacement is zero, which can be critical for stability analysis.
The expression g 2 − 36 can be factored using the difference of squares formula, leading to the result ( g − 6 ) ( g + 6 ) . This formula is applied by recognizing that g 2 and 36 are perfect squares. Thus, the final factored form is ( g − 6 ) ( g + 6 ) .
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