document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=771214>Question 771214</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, is anybody able to explain:<BR>\n" );
document.write( "WHY solutions of the quadratic equation y=7x^2-3x-11 (has two solutions because the discriminant is positive) are irrational?<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 #470031 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=470031\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The possibility of irrational solutions when y=0 is because x has been squared.  The solution process must take a step from a squared term, yielding the square root of the square expression.\r<BR>\n" );
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document.write( "You might start with something represented like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bp%29%5E2=V ALIGN=MIDDLE  ALT=\"%28x%2Bp%29%5E2=V\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and you want to know what is x.  Raise both sides to the power of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bp=%2Bsqrt%28V%29 ALIGN=MIDDLE  ALT=\"x%2Bp=%2Bsqrt%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bp=-sqrt%28V%29 ALIGN=MIDDLE  ALT=\"x%2Bp=-sqrt%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You need both the plus and the minus because either, when or if squared again, will become <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bp%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bp%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in its positive form (without showing any negative sign with x+p).\r<BR>\n" );
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document.write( "Continuing from there,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-p%2Bsqrt%28V%29 ALIGN=MIDDLE  ALT=\"x=-p%2Bsqrt%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-p-sqrt%28V%29 ALIGN=MIDDLE  ALT=\"x=-p-sqrt%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "V can possibly be not a perfect square, so x will be the two irrational solutions.  In case V is actually a perfect square, then naturally, the solutions for x would be rational.  \r<BR>\n" );
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document.write( "At the middle of your question, you showed the example equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7x%5E2-3x-11 ALIGN=MIDDLE  ALT=\"y=7x%5E2-3x-11\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Try setting y=0 and complete the square to solve for x.  You may see how this works and that you will have two irrational solutions for x.  We can check the discriminant now, yet before you try to solve for x:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%5E2-4%2A%287%29%2A%28-11%29=9%2B7%2A44=317 ALIGN=MIDDLE  ALT=\"%28-3%29%5E2-4%2A%287%29%2A%28-11%29=9%2B7%2A44=317\"   DISABLED_style= \"max-width:90%; height:auto;\" >, A POSITIVE VALUE.  TWO REAL SOLUTIONS, and because 317 is NOT a perfect square,  the solutions for x will be irrational. </SPAN>\n" );
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