document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=179914>Question 179914</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with this assignment!  \r<BR>\n" );
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document.write( "1.	Determine whether the following equations have a solution or not? Justify your answer. \r<BR>\n" );
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document.write( "a)	x2 + 3x - 15 = 0 <BR>\n" );
document.write( "b)	x2 + x + 4 = 0<BR>\n" );
document.write( "c)	 x2 – 4x - 7  = 0 <BR>\n" );
document.write( "d)	x2 – 8x + 16 = 0 <BR>\n" );
document.write( "e)	2x2 - 3x + 7 = 0 <BR>\n" );
document.write( "f)	x2 – 4x - 77 = 0 <BR>\n" );
document.write( "g)	3x2 - 7x + 6 = 0<BR>\n" );
document.write( "h)	4x2 + 16x + 16 = 0 \r<BR>\n" );
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document.write( "2.	If x = -3 and x = 4, then form a quadratic equation. \r<BR>\n" );
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document.write( "3.	What type of solution do you get for quadratic equations where D < 0? Give reasons for your answer. Also provide an example of such a quadratic equation and find the solution of the equation. \r<BR>\n" );
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document.write( "4.	Create a real-life situation that fits into the equation (x + 3)(x - 5) = 0 and express the situation as the same equation. \r<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 #134905 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134905\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "For problem #1, the answer depends on the set of numbers from which you will allow solutions.  They ALL have solutions over the Complex numbers, but some of them do not have solutions if you are only considering Real numbers.  All quadratic equations have 2 roots, all cubic equations have 3, and all nth degree polynomial equations have n roots.  If n is odd, one of the roots is guaranteed to be a real number because complex roots always occur in conjugate pairs.\r<BR>\n" );
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document.write( "To determine which of them have solutions over the Reals, compute the discriminant for each one.  For <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2 + bx + c = 0\">, the discriminant, <i>&#916;</i>, is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta = b^2 - 4ac\">.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta > 0\"> then the two roots are real and unequal.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta = 0\"> then the two roots are real and equal, sometimes referred to as a single root with a multiplicity of two.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta < 0\"> then there are no real number roots.  The two roots are a conjugate pair of complex numbers of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a \pm bi\"> where <i>i</i> is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i^2 = -1\">.\r<BR>\n" );
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document.write( "Problem 2:\r<BR>\n" );
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document.write( "Work backwards.  When you solve a quadratic by factoring, you reach a point where you have something that looks like <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x - a)(x - b) = 0\"> and then you apply the Zero Product Rule to conclude that <i>x</i>=<i>a</i> or <i>x</i>=<i>b</i>.  Beginning with <i>x</i>=<i>-3</i> or <i>x</i>=<i>4</i>, the factored quadratic must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 3)(x - 4) = 0\">.  Now all you need to do is multiply the two binomials using FOIL to get your desired quadratic equation.\r<BR>\n" );
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document.write( "Problem 3:  I already answered this in my response to Problem 1.  Just pick one of the examples in Problem 1 where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta < 0\"> and then use the quadratic formula,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{         } x_{1,2} = \frac {-b \pm sqrt{b^2-4ac}}{2a}\">, \r<BR>\n" );
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document.write( "to find the solutions.  Hint:  For the negative radicand, change the sign to plus and put an <i>i</i> in front of the radical.  That's because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large sqrt{-p} = sqrt{(-1)p} = i sqrt{p}\text { }(\math \forall p > 0)\"> \r<BR>\n" );
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document.write( "Problem 4:  John is one year older than his twin brothers, Tom and Jim.  The product of Tom and Jim's ages is 16.  How old is John?\r<BR>\n" );
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document.write( "John is <i>x</i> years old, so each of the twins is <i>x</i>-1 years old.  The product of their ages is then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x - 1)(x - 1) = 16\"> so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 - 2x + 1 = 16\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 - 2x - 15 = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 3)(x - 5) = 0\">\r<BR>\n" );
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document.write( "Throw out the <i>x</i>= -3 answer because no one can be -3 years old, and you are left with John is 5 years old.\r<BR>\n" );
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document.write( "Actually, my answer to 4 is sort of lame.  Maybe you can think of something better.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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