document.write( "Question 1195204: Solve these two equations:\r
\n" ); document.write( "\n" ); document.write( "(a) The quadratic equation $\"2ax%5E2-4ax%2Ba%2B1=0\"$ has two roots. If one root is five times the other, what is the value of a?\r
\n" ); document.write( "\n" ); document.write( "(b) If $\"x%2B2\"$ and $\"x-3\"$ are factors of the polynomial $\"p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb\"$ , find $\"a\"$ . \n" ); document.write( "

Algebra.Com's Answer #827618 by ikleyn(52817) $\"\"$ $\"About$

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\n" ); document.write( "Solve these two equations:\r
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\n" ); document.write( "\n" ); document.write( "(a)   The quadratic equation $\"2ax%5E2-4ax%2Ba%2B1=0\"$ has two roots.
\n" ); document.write( "        If one root is five times the other, what is the value of a?\r
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\n" ); document.write( "\n" ); document.write( "(b)   If $\"x%2B2\"$ and $\"x-3\"$ are factors of the polynomial $\"p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb\"$ ,
\n" ); document.write( "        find $\"a\"$ .
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Part (a)

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document.write( "Let z be one root of the equation; then 5z is another root.\r\n" );
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document.write( "Apply the Vieta's theorem.  it gives you these two equations for unknowns \"a\" and \"z\":\r\n" );
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document.write( "    (1)  the sum of the roots z and 5z is equal to  = 2\r\n" );
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document.write( "             z + 5z = 2,  or  6z = 2,  z =  = .\r\n" );
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document.write( "    (2)  the product of the roots is equal to \r\n" );
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document.write( "             z*(5z) = ,  which implies   = ,\r\n" );
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document.write( "              = ,  5*(2a) = 9*(a+1),  10a = 9a + 9,  a = 9.\r\n" );
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document.write( "Thus we get the ANSWER :  a = 9.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part (a) is solved.\r
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\n" ); document.write( "\n" ); document.write( "Two problems in your post are different.\r
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\n" ); document.write( "\n" ); document.write( "At this forum, I do not recommend to pack more than one problem to post, in order for do not create a MESS.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, post your second problem separately, and I will show you how to solve it.\r
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\n" ); document.write( "\n" ); document.write( "By the way, the rules of this forum PROHIBIT posting more than one problem at once.\r
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\n" ); document.write( "\n" ); document.write( "The rules are at this web-page\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/tutors/students/ask.mpl?action=ask_question&topic=Equations&return_url=http://www.algebra.com/algebra/homework/equations/\r
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\n" ); document.write( "\n" ); document.write( "from which you post your problems.\r
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\n" ); document.write( "\n" ); document.write( "It is assumed that you read these rules before posting.\r
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\n" ); document.write( "\n" ); document.write( "It is also assumed that you do understand what is written in that page and follow the rules.\r
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\n" ); document.write( "\n" ); document.write( "Those who violate them, work against their own interests.\r
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