document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209885>Question 1209885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f be the piecewise function such that<BR>\n" );
document.write( "f(x) =<BR>\n" );
document.write( "x^2 - 5x - 64 & \text{if} \ x \le 0, \\<BR>\n" );
document.write( "x^2 + 3x - 38 & \text{if} \ x > 0.\r<BR>\n" );
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document.write( "Find all x such that f(x) = 50.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850959 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850959\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are given the piecewise function:\r<BR>\n" );
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document.write( "$$ f(x) = \begin{cases} x^2 - 5x - 64 & \text{if } x \le 0 \\ x^2 + 3x - 38 & \text{if } x > 0 \end{cases} $$\r<BR>\n" );
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document.write( "We want to find all $x$ such that $f(x) = 50$.\r<BR>\n" );
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document.write( "**Case 1: $x \le 0$**\r<BR>\n" );
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document.write( "We have $x^2 - 5x - 64 = 50$, which simplifies to $x^2 - 5x - 114 = 0$.<BR>\n" );
document.write( "We can use the quadratic formula to solve for $x$:<BR>\n" );
document.write( "$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$<BR>\n" );
document.write( "In this case, $a = 1$, $b = -5$, and $c = -114$.<BR>\n" );
document.write( "$$ x = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(-114)}}{2(1)} = \frac{5 \pm \sqrt{25 + 456}}{2} = \frac{5 \pm \sqrt{481}}{2} $$<BR>\n" );
document.write( "We have two possible solutions:<BR>\n" );
document.write( "$$ x_1 = \frac{5 + \sqrt{481}}{2} \approx \frac{5 + 21.93}{2} \approx 13.465 $$<BR>\n" );
document.write( "$$ x_2 = \frac{5 - \sqrt{481}}{2} \approx \frac{5 - 21.93}{2} \approx -8.465 $$<BR>\n" );
document.write( "Since we are considering $x \le 0$, only $x_2$ is a valid solution.<BR>\n" );
document.write( "Thus, $x = \frac{5 - \sqrt{481}}{2}$ is a solution.\r<BR>\n" );
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document.write( "**Case 2: $x > 0$**\r<BR>\n" );
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document.write( "We have $x^2 + 3x - 38 = 50$, which simplifies to $x^2 + 3x - 88 = 0$.<BR>\n" );
document.write( "We can factor the quadratic as $(x - 8)(x + 11) = 0$.<BR>\n" );
document.write( "The solutions are $x = 8$ and $x = -11$.<BR>\n" );
document.write( "Since we are considering $x > 0$, only $x = 8$ is a valid solution.<BR>\n" );
document.write( "Thus, $x = 8$ is a solution.\r<BR>\n" );
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document.write( "Therefore, the solutions are $x = \frac{5 - \sqrt{481}}{2}$ and $x = 8$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{8, \frac{5 - \sqrt{481}}{2}}$<BR>\n" );
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