Question 1168999
the general formula for difference of squares is:


a^2 - b^2 = (a - b) * (a + b)
 

a is the square root of a^2 and b is the square root of b^2.


because of that, the general form for the equation of the difference of squares could be written as:


a^2 - b^2 = (sqrt(a^2) - sqrt(b^2)) * (sqrt(a^2) + sqrt(b^2))


since the square root of b^2 is equal to b, then it could be written as:


a^2 - b^2 = (sqrt(a^2) - b) * (sqrt(a^2) + b)


in this problem, b is left as is and a is replaced by sqrt(a^2 - 14a + 49), so the equation becomes:


sqrt(a^2 - 14a + 49))^2 - b^2 = (sqrt(a^2 - 14a + 49) - b) * (sqrt(a^2 - 14a + 49) + b)


what you need to do is find the square root of a^2 - 14a + 49.


fortunately, sqrt(a^2 - 14a + 49) is equal to a-7, because (a-7)^2 = a^2 - 14a + 49.


the equation of:


sqrt(a^2 - 14a + 49))^2 - b^2 = (sqrt(a^2 - 14a + 49) - b) * (sqrt(a^2 - 14a + 49) + b) becomes:


(a - 7)^2 - b^2 = (a - 7 + b) * (a - 7 - b)


to confirm this is accurate, you would perform the indicated operations to get:


(a - 7 + b) * (a - 7 - b) equals:


a * (a - 7 - b) - 7 * (a - 7 - b) + b * (a - 7 - b) which equals:


a^2 - 7a - ab -7a + 49 + 7b + ab - 7b - b^2.


group like terms together to get:


a^2 - 7a - 7a - ab + ab + 49 + 7b - 7b - b^2


combine like terms to get:


a^2 - 14a + 49 - b^2


that's the same as your original expression, so we're good.


your solution is:


a^2 - 14a + 49 - b^2 is equivalent to:


(a - 7 + b) * (a - 7 - b)