document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=173956>Question 173956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is one main question with several parts. Please help. Thank you,<BR>\n" );
document.write( "How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #128850 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=128850\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I don't think you stated your initial question precisely.  I think what you meant to ask is \"How do you know if a quadratic equation will have one, two, or no <i><b>real number</b></i> solutions?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In fact, the fundamental theorem of algebra tells us that a polynomial of degree n has n roots.  Therefore a quadratic, which is a polynomial of degree 2, must ALWAYS have two roots.  However, sometimes the roots are complex numbers so when you restrict your definition of 'solution' to the real numbers, you can have a 'no solution' result.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As to the question about one root, that is still a subject of some debate.  Some say that a quadratic always has two roots, but in the case of a perfect square trinomial, those two roots will be identical.  Others say that there is a single root with a multiplicity of two for that situation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The process to determine the character of the roots of a quadratic is as follows:\r<BR>\n" );
document.write( "\n" );
document.write( "Step 1:  Put the quadratic into standard form, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 2:  Calculate the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the expression under the radical in the quadratic formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, namely:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA=b%5E2-4ac ALIGN=MIDDLE  ALT=\"DELTA=b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA+=+0 ALIGN=MIDDLE  ALT=\"DELTA+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then you either have two real and identical roots, or you have one root with a multiplicity of two, depending on your school of thought.  This is your 'one solution' situation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA+%3E+0 ALIGN=MIDDLE  ALT=\"DELTA+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then you have two different real number roots.  This is your 'two solutions' situation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA+%3C+0 ALIGN=MIDDLE  ALT=\"DELTA+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then you have a conjugate pair of complex number roots of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B-bi ALIGN=MIDDLE  ALT=\"a%2B-bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2=-1 ALIGN=MIDDLE  ALT=\"i%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This is your 'no solution' situation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "***********************\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you can derive a quadratic polynomial with those two numbers as roots by using the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a root of a polynomial in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the polynomial.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the two roots of a quadratic, one of the quadratic polynomials with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > as roots can be derived from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-p%29%28x-q%29=x%5E2-%28p%2Bq%29x%2Bpq ALIGN=MIDDLE  ALT=\"%28x-p%29%28x-q%29=x%5E2-%28p%2Bq%29x%2Bpq\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "***********************\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your last question, \"Is it possible to have different quadratic equations with the same solution?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You need to be very careful with your language here.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-6=0 ALIGN=MIDDLE  ALT=\"x%5E2-x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a solution set {-2,3}, but so does <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2Bx%2B6=0 ALIGN=MIDDLE  ALT=\"-x%5E2%2Bx%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  However, one could successfully argue that these two equations are equivalent, and therefore are not really different quadratic equations.  So, the answer to your question as stated is no, you can't have different quadradic EQUATIONS with the same solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But do not confuse a quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a polynomial function of degree 2:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Looking at the previous example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-x-6 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%5E2%2Bx%2B6 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E2%2Bx%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > are two very different functions -- one is a parabola opening upwards and the other is a parabola opening downwards.   </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );