document.write( "Question 258319: A parabola y = ax^2 + bx + c has vertex (4, 2). If (2, 0) is on the parabola, then find the value of abc \n" ); document.write( "

Algebra.Com's Answer #190200 by Fombitz(32388) $\"\"$ $\"About$

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The formula for the x position of the vertex is
\n" ); document.write( " $\"x=-b%2F2a\"$
\n" ); document.write( " $\"-b%2F2a=4\"$
\n" ); document.write( " $\"b=-8a\"$
\n" ); document.write( "Now using the points,
\n" ); document.write( " $\"y+=+ax%5E2+%2B+bx+%2B+c+\"$
\n" ); document.write( " $\"2+=+16a+%2B+4b+%2B+c+\"$
\n" ); document.write( " $\"16a%2B4b%2Bc=2\"$
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\n" ); document.write( " $\"y+=+ax%5E2+%2B+bx+%2B+c+\"$
\n" ); document.write( " $\"0+=+4a+%2B+2b+%2B+c+\"$
\n" ); document.write( " $\"4a%2B2b%2Bc=0\"$
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\n" ); document.write( "Now substitute for b.
\n" ); document.write( " $\"b=-8a\"$
\n" ); document.write( " $\"16a%2B4b%2Bc=2\"$
\n" ); document.write( " $\"16a-32a%2Bc=2\"$
\n" ); document.write( "1. $\"-16a%2Bc=2\"$
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\n" ); document.write( " $\"4a%2B2b%2Bc=0\"$
\n" ); document.write( " $\"4a-16a%2Bc=0\"$
\n" ); document.write( "2. $\"-12a%2Bc=0%7D%7D%0D%0A%7B%7B%7Bc=12a\"$
\n" ); document.write( "Substitute into eq. 1 to solve for a.
\n" ); document.write( " $\"-16a%2Bc=2\"$
\n" ); document.write( " $\"-16a%2B12a=2\"$
\n" ); document.write( " $\"-4a=2\"$
\n" ); document.write( " $\"a=-1%2F2\"$
\n" ); document.write( "Then,
\n" ); document.write( " $\"c=12a\"$
\n" ); document.write( " $\"c=-6\"$
\n" ); document.write( "and finally,
\n" ); document.write( " $\"b=-8a\"$
\n" ); document.write( " $\"b=4\"$
\n" ); document.write( "So then,
\n" ); document.write( " $\"y+=+%28-1%2F2%29x%5E2+%2B+4x+-6+\"$
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\n" ); document.write( " $\"abc=%28-1%2F2%29%284%29%28-6%29=12\"$ \n" ); document.write( "

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