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Given , compute the sum of all the possible values of $\frac{14x+22}{x+2}=-3x+1$ Express your answer as a fraction.

Apr 25, 2021

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$$\frac{14x + 22}{x + 2} = -3x + 1$$

$$14x + 22 = (-3x + 1)(x + 2)$$

$$14x + 22 = -3x^2 -6x + x + 2$$

$$3x^2 + 6x + 14x - x + 22 - 2 = 0$$

$$3x^2 + 19x + 20 = 0$$

$$x = {-(19) \pm \sqrt{(19)^2-4(3)(20)} \over 2(3)}$$

$$x = {-19 \pm \sqrt{361-240} \over 6}$$

$$x = {-19 \pm \sqrt{121} \over 6}$$

$$x = {-19 \pm 11 \over 6}$$

$$x = -5$$

$$x = -\frac{4}{3}$$

$$-5 + -\frac{4}{3} = \frac{-15 - 4}{3} = -\frac{19}{3}$$

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Apr 25, 2021