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You listed this problem under \"Quadratic Equation\" so I assume the problem is to find a quadratic equation for the given points.
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\n" ); document.write( "\n" ); document.write( "The basic form of a quadratic equation is
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\n" ); document.write( "One way to find the equation is to substitute the coordinates of these points into the basic form. If we take the point (1, 1):
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\n" ); document.write( "1 = a + b + c
\n" ); document.write( "If we use the point (2, 3):
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\n" ); document.write( "3 = 4a + 2b + c
\n" ); document.write( "If we use the point (3, 6):
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\n" ); document.write( "6 = 9a + 3b + c
\n" ); document.write( "At this point we have a system of three equations with three variables (a, b and c). We should be able to solve this for a, b and c. The system is:
\n" ); document.write( "1 = a + b + c
\n" ); document.write( "3 = 4a + 2b + c
\n" ); document.write( "6 = 9a + 3b + c
\n" ); document.write( "There are a variety of ways to solve this system: Substitution, Linear Combination (aka Elimination), Cramer's Rule (determinants), and a variety of matrix-based methods. I will use Linear Combination. To start I will subtract the first equation from each of the other two equations. The result of these two subtractions:
\n" ); document.write( "2 = 3a + b
\n" ); document.write( "5 = 8a + 2b
\n" ); document.write( "Now I have a system of two equations of two variables (since the c's have been eliminated).
\n" ); document.write( "Now I will subtract twice the first equation from the second. The result of this subtraction:
\n" ); document.write( "1 = 2a
\n" ); document.write( "Dividing by two we get:
\n" ); document.write( "1/2 = a
\n" ); document.write( "Now that we have \"a\" we can substitute for \"a\" and find \"b\" and \"c\". Substituting for a in the second equation of the second system:
\n" ); document.write( "5 = 8(1/2) + 2b
\n" ); document.write( "5 = 4 + 2b
\n" ); document.write( "1 = 2b
\n" ); document.write( "1/2 = b
\n" ); document.write( "Now we can go back to the first system to find \"c\". Substituting in \"a\" and \"b\" into the second equation of the first system:
\n" ); document.write( "3 = 4(1/2) + 2(1/2) + c
\n" ); document.write( "3 = 2 + 1 + c
\n" ); document.write( "3 = 3 + c
\n" ); document.write( "0 = c
\n" ); document.write( "Now that we have \"a\", \"b\" and \"c\" we can write our quadratic equation by substituting these values into the basic form:
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\n" ); document.write( "Substituting:
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\n" ); document.write( "or
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\n" ); document.write( "Not only does this equation work for the three points we used, it also works for the fourth point, (4,10). \n" ); document.write( "