Factor [tex]9 x^2-16[/tex]
Answer (2)
Recognize the expression as a difference of squares: 9 x 2 − 16 = ( 3 x ) 2 − ( 4 ) 2 .
Apply the difference of squares factorization: a 2 − b 2 = ( a + b ) ( a − b ) .
Substitute a = 3 x and b = 4 to get ( 3 x + 4 ) ( 3 x − 4 ) .
The factored form of the expression is ( 3 x + 4 ) ( 3 x − 4 ) .
Explanation
Recognizing the Problem We are asked to factor the expression 9 x 2 − 16 . This expression is a difference of two squares.
Applying Difference of Squares We can rewrite the expression as ( 3 x ) 2 − ( 4 ) 2 . The difference of squares factorization is given by a 2 − b 2 = ( a + b ) ( a − b ) .
Factoring the Expression In our case, a = 3 x and b = 4 . Substituting these values into the difference of squares factorization, we get ( 3 x + 4 ) ( 3 x − 4 ) .
Selecting the Correct Option Comparing our result with the given options, we see that the correct factorization is ( 3 x + 4 ) ( 3 x − 4 ) .
Examples
Factoring the difference of squares is a useful technique in many areas of mathematics and physics. For example, when analyzing the motion of a projectile, you might encounter an expression like v 2 − u 2 , where v is the final velocity and u is the initial velocity. Factoring this as ( v + u ) ( v − u ) can simplify calculations and provide insights into the change in kinetic energy. Similarly, in electrical engineering, you might use difference of squares to simplify expressions involving impedances or power calculations. Understanding this factorization helps in simplifying complex problems and finding solutions more efficiently.
The expression 9 x 2 − 16 can be factored as ( 3 x + 4 ) ( 3 x − 4 ) , which is a result of recognizing it as a difference of squares. We apply the formula for factoring difference of squares to arrive at this result. Therefore, the factored form is ( 3 x + 4 ) ( 3 x − 4 ) .
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