Factor $36 x^2-49$
A. $(6 x+7)(-6 x-7)$
B. $(-6 x+7)(6 x-7)$
C. $(6 x+7)(6 x-7)$
D. $(6 x-7)^2$
Answer (1)
The expression 36 x 2 − 49 is a difference of squares. We can factor it as follows:
Recognize the difference of squares pattern: a 2 − b 2 = ( a + b ) ( a − b ) .
Identify a = 6 x and b = 7 .
Apply the factorization: ( 6 x + 7 ) ( 6 x − 7 ) .
Thus, the factored expression is ( 6 x + 7 ) ( 6 x − 7 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 36 x 2 − 49 . This looks like a difference of squares, which has the form a 2 − b 2 .
Rewriting the Expression We can rewrite 36 x 2 as ( 6 x ) 2 and 49 as 7 2 . So, we have ( 6 x ) 2 − 7 2 .
Applying the Difference of Squares The difference of squares factorization is a 2 − b 2 = ( a + b ) ( a − b ) . In our case, a = 6 x and b = 7 . Therefore, we have ( 6 x ) 2 − 7 2 = ( 6 x + 7 ) ( 6 x − 7 ) .
Final Factorization Thus, the factored form of 36 x 2 − 49 is ( 6 x + 7 ) ( 6 x − 7 ) .
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 involving the difference of squares when calculating the range of the projectile. Suppose the initial velocity is v and the angle of projection is θ . The range R can be expressed as R = g v 2 s i n ( 2 θ ) , where g is the acceleration due to gravity. If you want to find the angles for which the projectile has a specific range, you might need to factor expressions involving trigonometric functions, and recognizing the difference of squares can simplify the calculations.
