document.write( "Question 716254: Find the equation of the quadratic function with roots -8 and -6, \"a\" less than zero, and a vertex at (-7, 2). \n" ); document.write( "

Algebra.Com's Answer #439793 by jsmallt9(3758) $\"\"$ $\"About$

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When a number is a root of a polynomial function then (x - that number) is a factor of that function. So the quadratic function you are looking for, in factored form, will be:
\n" ); document.write( " $\"f%28x%29+=+a%28x-%28-8%29%29%28x-%28-6%29%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"f%28x%29+=+a%28x%2B8%29%28x%2B6%29\"$
\n" ); document.write( "In general, the \"a\" can be any non-zero number. But the problem states that \"a\" should be less than zero (or negative) and that we must have a vertex at (-7, 2). So there will probably be only one value for \"a\" that will fit. First we will multiply out the factors. Using FOIL on the last two factors:
\n" ); document.write( " $\"f%28x%29+=+a%28x%2Ax%2Bx%2A6%2B8%2Ax%2B8%2A6%29\"$
\n" ); document.write( " $\"f%28x%29+=+a%28x%5E2%2B6x%2B8x%2B48%29\"$
\n" ); document.write( " $\"f%28x%29+=+a%28x%5E2%2B14x%2B48%29\"$
\n" ); document.write( "Distributing the \"a\":
\n" ); document.write( " $\"f%28x%29+=+ax%5E2%2B14ax%2B48a%29\"$

\n" ); document.write( "Now we set out to figure out what \"a\" must be. The vertex, (-7, 2), should fit this equation and we can use this to find \"a\". Substituting -7 for x and 2 for y/f(x) we get:
\n" ); document.write( " $\"2+=+a%28-7%29%5E2%2B14a%28-7%29%2B48a%29\"$
\n" ); document.write( "Now we solve for \"a\". We start by simplifying...
\n" ); document.write( " $\"2+=+a%2849%29%2B14a%28-7%29%2B48a%29\"$
\n" ); document.write( " $\"2+=+49a%2B%28-98a%29%2B48a%29\"$
\n" ); document.write( " $\"2+=+-a%29\"$
\n" ); document.write( "Dividing by -1:
\n" ); document.write( " $\"-2+=+a\"$ \n" ); document.write( "

\n" );