document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192071>Question 1192071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the values of​ a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 4-√ 17 and 4+√ 17.<BR>\n" );
document.write( "​(Hint: Use the​ zero-product property in​ reverse.) thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823970 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=823970\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4+-+sqrt%2817%29 ALIGN=MIDDLE  ALT=\"x+=+4+-+sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4+%2B+sqrt%2817%29 ALIGN=MIDDLE  ALT=\"x+=+4+%2B+sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the two roots of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m-n ALIGN=MIDDLE  ALT=\"m-n\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bn ALIGN=MIDDLE  ALT=\"m%2Bn\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Where,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+4 ALIGN=MIDDLE  ALT=\"m+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+sqrt%2817%29 ALIGN=MIDDLE  ALT=\"n+=+sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use Vieta's formulas to sum the roots:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Broot2+=+%28m-n%29%2B%28m%2Bn%29 ALIGN=MIDDLE  ALT=\"root1%2Broot2+=+%28m-n%29%2B%28m%2Bn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Broot2+=+2m+ ALIGN=MIDDLE  ALT=\"root1%2Broot2+=+2m+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Broot2+=+8 ALIGN=MIDDLE  ALT=\"root1%2Broot2+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and multiply the roots as well<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Aroot2+=+%28m-n%29%2A%28m%2Bn%29+ ALIGN=MIDDLE  ALT=\"root1%2Aroot2+=+%28m-n%29%2A%28m%2Bn%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Aroot2+=+m%5E2+-+n%5E2 ALIGN=MIDDLE  ALT=\"root1%2Aroot2+=+m%5E2+-+n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Difference of squares rule<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Aroot2+=+4%5E2+-+%28sqrt%2817%29%29%5E2 ALIGN=MIDDLE  ALT=\"root1%2Aroot2+=+4%5E2+-+%28sqrt%2817%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Aroot2+=+16+-17 ALIGN=MIDDLE  ALT=\"root1%2Aroot2+=+16+-17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root1%2Aroot2+=+-1 ALIGN=MIDDLE  ALT=\"root1%2Aroot2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The negative of the sum of the roots is the value of b, while c represents the product of the roots.<BR>\n" );
document.write( "b = -8 and c = -1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Why does this work? Consider the two roots to be r and s<BR>\n" );
document.write( "This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+r ALIGN=MIDDLE  ALT=\"x+=+r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+s ALIGN=MIDDLE  ALT=\"x+=+s\"   DISABLED_style= \"max-width:90%; height:auto;\" > lead to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-r+=+0 ALIGN=MIDDLE  ALT=\"x-r+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+s+=+0 ALIGN=MIDDLE  ALT=\"x+-+s+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > respectively.<BR>\n" );
document.write( "Furthermore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-r%29%28x-s%29+=+x%5E2-sx-rx%2Brs+=+x%5E2-%28s%2Br%29x%2Brs+=+0 ALIGN=MIDDLE  ALT=\"%28x-r%29%28x-s%29+=+x%5E2-sx-rx%2Brs+=+x%5E2-%28s%2Br%29x%2Brs+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The -(s+r)x term matches with bx, showing that b = -(s+r). We add the two roots, then flip the sign to get the value of b.<BR>\n" );
document.write( "The rs term is the product of the roots and matches with c.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+8x+-+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+8x+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can confirm this answer by using the quadratic formula to arrive back to the original roots mentioned.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another approach:\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+=+4+-+sqrt%2817%29 ALIGN=MIDDLE  ALT=\"x+=+4+-+sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4+=+-sqrt%2817%29 ALIGN=MIDDLE  ALT=\"x-4+=+-sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+=+%28-sqrt%2817%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+%28-sqrt%2817%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+=+17 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Through very nearly identical steps, you should find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2Bsqrt%2817%29 ALIGN=MIDDLE  ALT=\"x+=+4%2Bsqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > also leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+=+17 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the square root method, solving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+=+17 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads back to the original roots mentioned.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's expand things out and get everything to one side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+=+17 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%28x-4%29+=+17 ALIGN=MIDDLE  ALT=\"%28x-4%29%28x-4%29+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-4%29-4%28x-4%29+=+17 ALIGN=MIDDLE  ALT=\"x%28x-4%29-4%28x-4%29+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x-4x%2B16+=+17 ALIGN=MIDDLE  ALT=\"x%5E2-4x-4x%2B16+=+17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x-4x%2B16-17+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-4x-4x%2B16-17+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x-1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-8x-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We get the same answer as before.<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );