document.write( "Question 629298: Find the quadratic polynomial whose graph goes through the points (−1,5), (0,5), and (2,29). \r
\n" ); document.write( "\n" ); document.write( "f(x)= _x2+_ x+_ <<< that form\r
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\n" ); document.write( "\n" ); document.write( "I got to the step 4a+2b+0=29\r
\n" ); document.write( "\n" ); document.write( "that was like my 6/7th step. I am getting stuck from there. Can i get any help please?
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Algebra.Com's Answer #852468 by ikleyn(52965) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Find the quadratic polynomial whose graph goes through the points (−1,5), (0,5), and (2,29).
\n" ); document.write( "f(x)= _x2+_ x+_ <<< that form\r
\n" ); document.write( "\n" ); document.write( "I got to the step 4a+2b+0=29
\n" ); document.write( "that was like my 6/7th step. I am getting stuck from there. Can i get any help please?
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\n" ); document.write( "\n" ); document.write( "        This task, as it is solved in the post by @Theo, seems routine and does not create any inspiration.\r
\n" ); document.write( "\n" ); document.write( "        But it depends on how to look at it.\r
\n" ); document.write( "\n" ); document.write( "        I will show you another solution from different perspective that will make it seem beautiful like a sparkling diamond.\r
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document.write( "Notice that the y-coordinate is the same, '5', for both points (-1,5) and (0,5).\r\n" );
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document.write( "It means that if you consider another quadratic function, g(x) = f(x) - 5, instead of f(x),\r\n" );
document.write( "this new quadratic function will have x-intercepts at  x= -1  and  x= 0.\r\n" );
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document.write( "In turn, it means that g(x) = ax*(x+1) with some real coefficient 'a',\r\n" );
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document.write( "so f(x) = ax(x+1) + 5.\r\n" );
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document.write( "Great (!)   So, to determine f(x), we only need to find single unknown coefficient 'a'.\r\n" );
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document.write( "For it, we use the condition that the parabola passes through the point (2,29).\r\n" );
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document.write( "It gives us this equation\r\n" );
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document.write( "    f(2) = 29,  or  a*2*(2+1) + 5 = 29,  or  2*3*a = 29 - 5,  or  6a = 24.\r\n" );
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document.write( "Thus we find  a = 24/6 = 4.\r\n" );
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document.write( "So, our quadratic function is  f(x) = 4x(x+1) + 5,  or  f(x) = 4x^2 + 4x + 5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, the problem is solved completely, and we obtained the solution
\n" ); document.write( "with minimal computations and maximal fun.\r
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\n" ); document.write( "\n" ); document.write( "        This trick works for many other similar problem, where \r
\n" ); document.write( "\n" ); document.write( "        two points of the three given points have the same y-coordinate.\r
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\n" ); document.write( "\n" ); document.write( "                So, this method is worth to learn ( ! ) \r
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