x x + 1 - 56 = 4 x x - 7 What is the sum of the solutions to

Question

x x + 1 - 56 = 4 x x - 7 What is the sum of the solutions to the given equation?

Accepted Answer

The correct answer is 29 3 . Applying the distributive property to the left-hand side of the given equation, x x + 1 - 56 , yields x 2 + x - 56 . Applying the distributive property to the right-hand side of the given equation, 4 x x - 7 , yields 4 x 2 - 28 x . Thus, the equation becomes x 2 + x - 56 = 4 x 2 - 28 x . Combining like terms on the left- and right-hand sides of this equation yields 0 = 4 x 2 - x 2 + - 28 x - x + 56 , or 3 x 2 - 29 x + 56 = 0 . For a quadratic equation in the form a x 2 + b x + c = 0 , where a , b , and c are constants, the quadratic formula gives the solutions to the equation in the form x = - b /mo> b 2 - 4 a c 2 a . Substituting 3 for a , - 29 for b , and 56 for c from the equation 3 x 2 - 29 x + 56 = 0 into the quadratic formula yields x = 29 /mo> - 29 2 - 4 3 56 2 3 , or x = 29 6 /mo> 13 6 . It follows that the solutions to the given equation are 29 6 + 13 6 and 29 6 - 13 6 . Adding these two solutions gives the sum of the solutions: 29 6 + 13 6 + 29 6 - 13 6 , which is equivalent to 29 6 + 29 6 , or 29 3 . Note that 29/3, 9.666, and 9.667 are examples of ways to enter a correct answer.

Question x-x-1-