The function g is defined by g x = x x - 2 x + 6 2 . The value

Question

The function g is defined by g x = x x - 2 x + 6 2 . The value of g ( 7 - w ) is 0 , where w is a constant. What is the sum of all possible values of w ?

Accepted Answer

The correct answer is 25 . The value of g 7 - w is the value of g x when x = 7 - w , where w is a constant. Substituting 7 - w for x in the given equation yields g 7 - w = 7 - w 7 - w - 2 7 - w + 6 2 , which is equivalent to g 7 - w = 7 - w 5 - w 13 - w 2 . It&rsquo;s given that the value of g 7 - w is 0 . Substituting 0 for g 7 - w in the equation g 7 - w = 7 - w 5 - w 13 - w 2 yields 0 = 7 - w 5 - w 13 - w 2 . Since the product of the three factors on the right-hand side of this equation is equal to 0 , at least one of these three factors must be equal to 0 . Therefore, the possible values of w can be found by setting each factor equal to 0 . Setting the first factor equal to 0 yields 7 - w = 0 . Adding w to both sides of this equation yields 7 = w . Therefore, 7 is one possible value of w . Setting the second factor equal to 0 yields 5 - w = 0 . Adding w to both sides of this equation yields 5 = w . Therefore, 5 is a second possible value of w . Setting the third factor equal to 0 yields 13 - w 2 = 0 . Taking the square root of both sides of this equation yields 13 - w = 0 . Adding w to both sides of this equation yields 13 = w . Therefore, 13 is a third possible value of w . Adding the three possible values of w yields 7 + 5 + 13 , or 25 . Therefore, the sum of all possible values of w is 25 .

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