What is $g^2-36$ in factored form?
Answer (2)
Recognize the expression as a difference of squares: g 2 − 36 .
Apply the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) .
Substitute a = g and b = 6 into the formula.
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 factoring pattern.
Applying the Difference of Squares The difference of squares pattern is a 2 − b 2 = ( a − b ) ( a + b ) . In our case, a = g and b = 6 , since 36 = 6 2 .
Factoring the Expression Therefore, we can factor g 2 − 36 as ( g − 6 ) ( g + 6 ) .
Examples
The difference of squares factorization is useful in many areas, such as simplifying algebraic expressions and solving equations. For example, if you are designing a rectangular garden and want to know the possible dimensions given a certain area that can be expressed as a difference of squares, factoring can help you find those dimensions. Suppose the area of the garden is given by x 2 − 9 , where x is some variable related to the dimensions. By factoring this expression into ( x − 3 ) ( x + 3 ) , you can determine possible lengths and widths for the garden, which is a practical application of this factoring technique.
The expression g 2 − 36 can be factored as ( g − 6 ) ( g + 6 ) using the difference of squares method. This method applies when you have a square number subtracted from another square number. Therefore, the final factorized expression is ( g − 6 ) ( g + 6 ) .
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