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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$5 (x + 7) = 15 (x - 17) (x + 7)$

What is the sum of the solutions to the given equation?

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Explanation

The correct answer is $StartFraction 31 Over 3 EndFraction$ . Subtracting $5 left parenthesis x plus 7 right parenthesis$ from each side of the given equation yields $0 = 15 (x - 17) (x + 7) - 5 (x + 7)$ . Since $5 left parenthesis x plus 7 right parenthesis$ is a common factor of each of the terms on the right-hand side of this equation, it can be rewritten as $0 = 5 (x + 7) (3 (x - 17) - 1)$ . This is equivalent to $0 = 5 (x + 7) (3 x - 51 - 1)$ , or $0 = 5 (x + 7) (3 x - 52)$ . Dividing both sides of this equation by $5$ yields $0 = (x + 7) (3 x - 52)$ . Since a product of two factors is equal to $0$ if and only if at least one of the factors is $0$ , either $x + 7 = 0$ or $3 x - 52 = 0$ . Subtracting $7$ from both sides of the equation $x + 7 = 0$ yields $x = - 7$ . Adding $52$ to both sides of the equation $3 x - 52 = 0$ yields $3 x equals 52$ . Dividing both sides of this equation by $3$ yields $x = \frac{52}{3}$ . Therefore, the solutions to the given equation are $negative 7$ and $StartFraction 52 Over 3 EndFraction$ . It follows that the sum of the solutions to the given equation is $- 7 + \frac{52}{3}$ , which is equivalent to $- \frac{21}{3} + \frac{52}{3}$ , or $StartFraction 31 Over 3 EndFraction$ . Note that 31/3 and 10.33 are examples of ways to enter a correct answer.