Multiply and add like terms: $(5 z+6)(5 z-6)$
Answer (2)
Recognize the expression as a difference of squares: ( a + b ) ( a − b ) = a 2 − b 2 .
Apply the difference of squares formula with a = 5 z and b = 6 .
Calculate ( 5 z ) 2 = 25 z 2 and 6 2 = 36 .
Write the simplified expression: 25 z 2 − 36 .
Explanation
Recognizing the Pattern We are asked to multiply and simplify the expression ( 5 z + 6 ) ( 5 z − 6 ) . This looks like a special product called the difference of squares.
Applying the Difference of Squares The difference of squares formula is ( a + b ) ( a − b ) = a 2 − b 2 . In our case, a = 5 z and b = 6 .
Substitution Now, we substitute a and b into the formula: ( 5 z + 6 ) ( 5 z − 6 ) = ( 5 z ) 2 − ( 6 ) 2
Calculating the Squares Next, we calculate the squares: ( 5 z ) 2 = 5 2 ⋅ z 2 = 25 z 2 6 2 = 36
Simplified Expression Finally, we write the simplified expression: 25 z 2 − 36
Examples
The difference of squares pattern is useful in various fields. For example, in engineering, it can simplify calculations related to areas and volumes. Imagine you are designing a square garden with side length 5 z meters and you want to remove a square section of side length 6 meters. The remaining area can be calculated using the difference of squares: ( 5 z ) 2 − 6 2 = ( 5 z + 6 ) ( 5 z − 6 ) . This simplifies to 25 z 2 − 36 square meters, making it easier to determine the amount of soil or fencing needed for the remaining area.
The expression ( 5 z + 6 ) ( 5 z − 6 ) can be simplified using the difference of squares formula, resulting in 25 z 2 − 36 . By recognizing the structure of the expression, we can avoid lengthy calculations. The final simplified expression is 25 z 2 − 36 .
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