document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=535981>Question 535981</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One solution of kx^2-5x+k=0 is 3. Find the other solution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352095 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=352095\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Problem: if one solution of the quadratic equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kx%5E2+-+5x+%2B+k+=+0 ALIGN=MIDDLE  ALT=\"kx%5E2+-+5x+%2B+k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "is 3, what is the other solution?<BR>\n" );
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document.write( "Notice that when you term-by-term compare the given equation to the standard form of the quadratic equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "you can see that \"a\" correlates to k, \"b\" equals -5, and \"c\" also equals k.<BR>\n" );
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document.write( "We know that the quadratic formula applies to solving a quadratic equation in the standard form. The quadratic formula says that for a quadratic equation in the standard form x can be found from:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "into this formula we can substitute k for \"a\" and \"c\" and -5 for \"b\" to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2Ak%2Ak+%29%29%2F%282%2Ak%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2Ak%2Ak+%29%29%2F%282%2Ak%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-5%29 ALIGN=MIDDLE  ALT=\"-%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes + 5 and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5%29%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28-5%29%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2Ak+=+k%5E2 ALIGN=MIDDLE  ALT=\"k%2Ak+=+k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. These actions result in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+%2B-+sqrt%28+25-4%2Ak%5E2+%29%29%2F%282%2Ak%29+ ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%28+25-4%2Ak%5E2+%29%29%2F%282%2Ak%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "multiply both sides by 2*k which is the denominator on the right side. This clears the denominator, and the formula becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ak%2Ax+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+ ALIGN=MIDDLE  ALT=\"2%2Ak%2Ax+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "we also know that one value for x is 3. We were told that in the statement of the problem. So substitute 3 for x in the formula. This makes the left side become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ak%2A3+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+ ALIGN=MIDDLE  ALT=\"2%2Ak%2A3+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "multiply out the left side to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Ak+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+ ALIGN=MIDDLE  ALT=\"6%2Ak+=+5+%2B-+sqrt%28+25-4%2Ak%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "get rid of the 5 on the right side by subtracting 5 from both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Ak+-+5+=+0+%2B-+sqrt%28+25-4%2Ak%5E2+%29+ ALIGN=MIDDLE  ALT=\"6%2Ak+-+5+=+0+%2B-+sqrt%28+25-4%2Ak%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "square both sides to get rid of the radical on the right side and this formula becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2Ak%5E2+-+60k+%2B+25+=+25-4%2Ak%5E2++ ALIGN=MIDDLE  ALT=\"36%2Ak%5E2+-+60k+%2B+25+=+25-4%2Ak%5E2++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "cancel 25 on both sides of the formula by subtracting 25 from both sides. Also get rid of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Ak%5E2 ALIGN=MIDDLE  ALT=\"-4%2Ak%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ak%5E2 ALIGN=MIDDLE  ALT=\"4%2Ak%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%2Ak%5E2+-+60k+=+0 ALIGN=MIDDLE  ALT=\"40%2Ak%5E2+-+60k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factor a k from terms on the left side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2A%2840k+-+60%29+=+0 ALIGN=MIDDLE  ALT=\"k%2A%2840k+-+60%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "notice that this equation will be true if either of the two factors on the left side is equal to zero because multiplication by a zero results in the left side becoming zero and making the left side equal to the zero on the right side.<BR>\n" );
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document.write( "So this formula will be true if either k = 0 or if 40k - 60 = 0. We can ignore k = 0 because if this were true in the original quadratic equation of the problem if k were zero, both the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%7D%7D%5D+term+and+the+constant+term+would+be+zero%2C+making+the+original+quadratic+equation+reduce+to+-5x+=+0.+This+doesn%27t+make+sense+since+it+requires+that+x+can+only+equal+for+the+reduced+equation+to+be+true.%0D%0A.%0D%0AWe+are+therefore+left+with+40k+-+60+=+0%0D%0A.%0D%0ASolve+this+by+adding+60+to+both+sides%3A%0D%0A.%0D%0A%7B%7B%7B40k+=+60 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "then dividing both sides by 60:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+60%2F40 ALIGN=MIDDLE  ALT=\"k+=+60%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and reducing the right side by dividing both the numerator and denominator by 20 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+3%2F2 ALIGN=MIDDLE  ALT=\"k+=+3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we can go back to the original problem and substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for k to make it become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29x%5E2+-+5x+%2B+3%2F2++=+0 ALIGN=MIDDLE  ALT=\"%283%2F2%29x%5E2+-+5x+%2B+3%2F2++=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply all terms by 2 to cancel out the denominator and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+10x+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2+-+10x+%2B+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "compared to the standard quadratic equation we now have \"a\" = 3, \"b\" = -10, and \"c\" = 3. Substituting these values into the quadratic formula, we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2A3%2A3+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2A3%2A3+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplifying to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%28100-36%29%29%2F6 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%28100-36%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The radical term becomes the square root of 64:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+sqrt%2864%29%29%2F6 ALIGN=MIDDLE  ALT=\"x+=+%2810+%2B-+sqrt%2864%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810+%2B-+8%29%2F6%29%29%29%0D%0A.%0D%0Aso+the+two+answers+for+x+are%3A%0D%0A.%0D%0A%7B%7B%7Bx+=+18%2F6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and this simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3 ALIGN=MIDDLE  ALT=\"x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%2F6 ALIGN=MIDDLE  ALT=\"x+=+2%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F3 ALIGN=MIDDLE  ALT=\"x+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The first value for x we already knew because it was part of the problem. The second value for x, which we found was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F3 ALIGN=MIDDLE  ALT=\"x+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is the answer you were asked to find in the problem. <BR>\n" );
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document.write( "Hope this helps you to understand the quadratic equation a little better and how to apply that knowledge to solving this problem.<BR>\n" );
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document.write( "Check my work. It's late and I might have introduced some errors due to lack of coffee. You can also check this problem by multiplying the two factors:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29%2A%28x+-+3%29%2A%28x+-+1%2F3%29 ALIGN=MIDDLE  ALT=\"%283%2F2%29%2A%28x+-+3%29%2A%28x+-+1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and see if the answer does not give you the original equation in the problem in which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+3%2F2 ALIGN=MIDDLE  ALT=\"k+=+3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Good luck ...<BR>\n" );
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document.write( " </SPAN>\n" );
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