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# $f(x) = \left\{ \begin{array}{cl} 2x + 7 & \text{if } x < -2, \\ -x^2 - x + 1 & \text{if } x \ge -2. \end{array} \right.$

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Let$f(x) = \left\{ \begin{array}{cl} 2x + 7 & \text{if } x < -2, \\ -x^2 - x + 1 & \text{if } x \ge -2. \end{array} \right.$Find the sum of all values of $x$ such that $f(x) = -5.$

Sep 27, 2021

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2x+ 7 = -5   implies x = -6  since this is less than -2 it is a valid solution

-x^2 - x + 1 = -5  can be written as x^2 + x - 6 = 0  which factors as (x-2)(x+3)=0  for which the only solution that is greater than or equal to -2 is x = 2.

Hence the sum of the valid values of x is -6 + 2 = -4

Sep 27, 2021