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1.) Let\[f(x) = \left\{ \begin{array}{cl} 2x + 1 & \text{if } x \le 3, \\ 8 - 4x & \text{if } x > 3. \end{array} \right.\]Find the sum of all values of $x$ such that $f(x) = 0.$

 

2.) The graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$.

 Aug 23, 2020
 #1
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1. x = 14.

 

2. a + b = -1 + 8 = 7.

 Aug 23, 2020
 #2
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Sorry, but you answer is wrong as you've used some random numbers. The correct answer is below:

 

The vertical asymptotes will occur when the denominator of a simplified rational expression is equal to zero. We factor the denominator \(3x^2+5x+2\) to obtain \((3x + 2)(x + 1)\). Hence, there are vertical asymptotes when \(x=1,-2/3\), and the sum of these values of \(-1-2/3\) is \(-5/3\)

(We can also use Vieta's formulas, which states that the sum of the roots of \(ax^2+bx+c=0\) is \(-b/a\).)

 Aug 24, 2020

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