document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207789>Question 1207789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use your graphing calculator to graph the quadratic function f(x)=5x^2+6x-17 . <BR>\n" );
document.write( "Which statement about the roots is true? <BR>\n" );
document.write( "(a) There are no roots. <BR>\n" );
document.write( "(b) The positive root is between 0 and 1. <BR>\n" );
document.write( "(c) The positive root is between 3 and 2. <BR>\n" );
document.write( "(d) The negative root is between -3 and -2. </SPAN>\n" );
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document.write( "Answer: <font color=red>Choice (d)</font><BR>\n" );
document.write( "The negative root is between -3 and -2.\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Your graphing calculator will help you find the answer very quickly. <BR>\n" );
document.write( "There's not much for me to say in this regard. <BR>\n" );
document.write( "You should notice the parabola crosses through the x axis between x = -3 and x = -2 which leads to <font color=red>answer choice (d)</font><BR>\n" );
document.write( "The other root crosses between x = 1 and x = 2, but that option isn't available in the answer choices above.<BR>\n" );
document.write( "Desmos and GeoGebra are both free online calculators in case you don't have something like a TI83.<BR>\n" );
document.write( "There are many other options as well.\r<BR>\n" );
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document.write( "To make this question more interesting, let's suppose that your teacher may not want you to use a graphing calculator. <BR>\n" );
document.write( "Use a non-graphing calculator (or pencil & paper) to determine f(0) = -17 and f(1) = -6. <BR>\n" );
document.write( "I'll skip showing steps since it's basic arithmetic. Follow the order of operations PEMDAS. <BR>\n" );
document.write( "The results are both negative. There isn't a sign change so we cannot be certain there's a root between x = 0 and x = 1. <BR>\n" );
document.write( "There may be a root or there may not be. A root occurs when f(x) = 0. <BR>\n" );
document.write( "So we cannot say for certain that statement (b) is true or not. \r<BR>\n" );
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document.write( "Furthermore, f(3) = 46 is positive while f(2) = 15 is also positive. <BR>\n" );
document.write( "We don't know if there's a root between x = 2 and x = 3 since f(x) didn't change sign.<BR>\n" );
document.write( "This means we cannot be certain that statement (c) is true or not.\r<BR>\n" );
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document.write( "Lastly, f(-3) = 10 is positive which contrasts with f(-2) = -9 being negative. <BR>\n" );
document.write( "Finally we have a sign change. <BR>\n" );
document.write( "We are guaranteed <font color=red>at least one root between x = -3 and x = -2</font>. <BR>\n" );
document.write( "This leads us to <font color=red>answer choice (d)</font> which rules out choice (a).\r<BR>\n" );
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document.write( "Side notes:<ul><li>Use of the quadratic formula yields these approximate roots {x = <font color=red>-2.539</font>, x = 1.339} the first root listed confirms <font color=red>choice (d)</font> is the answer.</li><li>Notice how we don't care about the actual results of the functions. All we care about is if the result is positive or negative (since we're looking for sign changes in f(x) ).</li><li>For more info check out \"intermediate value theorem\". Specifically as it pertains to roots.</li></ul>Good luck with the rest of your homework.<BR>\n" );
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