document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=986986>Question 986986</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) Is y = x^2 + 7x + 12 a quadratic equation*? Why or why not?\r<BR>\n" );
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document.write( "b) Can it be solved by factoring? If yes, solve it. If not, why not?\r<BR>\n" );
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document.write( "c) As it happens, the equation y = x^2 + 3x - 23 also cannot be solved by factoring. One method of solving quadratic equations, however, that will solve ALL of them, is called using the quadratic formula. Can you state and explain the quadratic formula?\r<BR>\n" );
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document.write( "d) Use the quadratic formula to solve y = x^2 + 3x - 23 and explain your steps.\r<BR>\n" );
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document.write( "Thanks<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 #607764 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=607764\">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( "A quadratic equation is any equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\ +\ bx\ +\ c\ =\ 0\"> where  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ \not{=}\ 0\">.  For your purposes at this point in your study of mathematics, assume that all of the coefficients are real numbers.  However, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> is a quadratic <i>equation</i> if and only if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> is equal to some specific value such that with appropriate algebraic manipulation the equation can be  arranged in to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\ +\ bx\ +\ c\ =\ 0\"> form.  Without such a restriction on the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> is a quadratic <i>function</i>.\r<BR>\n" );
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document.write( "Quadratic <i>functions</i> cannot be \"solved\" in the commonly accepted sense of finding specific numbers that make the equation a true statement.  The only thing you can say about the solution set of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> is that it is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\{(x,y)\ \in\ \mathbb{R}^2\,:\,y\ =\ ax^2\ +\ bx\ +\ c\right\}\">.  Trivial in the extreme.  It is only when you specify a particular value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> that the function becomes a quadratic equation that is solvable for a particular pair of values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">\r<BR>\n" );
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document.write( "In fact, <i>all</i> quadratic equations can be reduced to the product of a pair of factors, although they may not necessarily be rational or even real.  However, unless the factors are rational, finding the factors by ordinary factorization means may be insurmountably challenging.  All of this to say that you stated this question incorrectly. What you are really being asked is if the quadratic equation in question is factorable <i>over the rationals</i>.  Fortunately, there is a simple way to tell:  The quadratic is factorable over the rationals if and only if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^2\ -\ 4ac\"> is a perfect square.  In the case of your example, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 49\ -\ 48\ =\ 1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\"> is a perfect square.  Therefore the quadratic trinomial <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ 7x\ +\ 12\"> is factorable over the rationals.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ 7x\ +\ 12\ =\ 0\"> can be solved by factoring. Hint: 3 + 4 = 7 and 3 * 4 = 12.\r<BR>\n" );
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document.write( "The quadratic formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-b\ \pm\ \sqrt{b^2\ -\ 4ac}}{2a}\">\r<BR>\n" );
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document.write( "Plug in your numbers and do the arithmetic.\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" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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