document.write( "Question 870120: I need help on this one please\r
\n" ); document.write( "\n" ); document.write( "5x^2 = 8x+21 \n" ); document.write( "
Algebra.Com's Answer #524712 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
\"5x%5E2+=+8x%2B21\" is a quadratic equation.
\n" ); document.write( "It would be customarily written in the more traditional form
\n" ); document.write( "\"5x%5E2+-+8x-21=0\" .
\n" ); document.write( "There are several ways to solve it.
\n" ); document.write( "All quadratic equations can be solved two ways:
\n" ); document.write( "by \"completing the square\", and
\n" ); document.write( "by using the quadratic formula.
\n" ); document.write( "Some quadratic equations can also be solved by factoring.
\n" ); document.write( "Each method has its advantages and disadvantages,
\n" ); document.write( "some equations make one method look easier,
\n" ); document.write( "and people have their preferences for one or another method.
\n" ); document.write( "
\n" ); document.write( "FACTORING sometimes is very easy.
\n" ); document.write( "It is not too easy for this equation, but it works, and it's what I would do.
\n" ); document.write( "The coefficients \"5\" and \"-21\" (at the ends) multiply to give \"5%2A%28-21%29=-105\".
\n" ); document.write( "I will look for pairs of factors of \"-105\" that add up to \"-8\" , which is the middle coefficient.
\n" ); document.write( "\"105=1%2A105=3%2A35=5%2A21=7%2A15\"
\n" ); document.write( "Since \"7%2A%2815%29=-105\" and \"7%2B%28-18%29=-8\" , \"7\" and \"-15\" will be the coefficients that I need.
\n" ); document.write( "I factor \"5x%5E2-8x-21\" \"by parts\" as follows:
\n" ); document.write( "
\n" ); document.write( "Then I re-write the equation \"5x%5E2-8x-21=0\" as
\n" ); document.write( "\"%285x%2B7%29%28x-3%29=0\" and realize that for the product to be zero, one factor must be zero,
\n" ); document.write( "so either \"x-3=0\"-->\"highlight%28x=3%29\" ,
\n" ); document.write( "or \"5x%2B7=0\"-->\"5x=-7\"-->\"highlight%28x=-7%2F5%29\"
\n" ); document.write( "
\n" ); document.write( "COMPLETING THE SQUARE sometimes is very easy, it always works, and does not require remembering formulas.
\n" ); document.write( "It is not too easy in this case either, but here it goes:
\n" ); document.write( "\"5x%5E2-8x-21=0\"-->\"5x%5E2-8x=21\"-->\"%285x%5E2-8x%29%2F5=21%2F5\"-->\"x%5E2-%288%2F5%29x=21%2F5\"
\n" ); document.write( "
\n" ); document.write( "so, from \"x%5E2-%288%2F5%29x=21%2F5\" ,
\n" ); document.write( "adding \"%284%2F5%29%5E2=16%2F25\" to both sides of the equal sign, I get the equivalent equation
\n" ); document.write( "\"x%5E2-%288%2F5%29x%2B%284%2F5%29%5E2=21%2F5%2B16%2F25\" , which I can re-write as
\n" ); document.write( "\"%28x-4%2F5%29%5E2=105%2F25%2B16%2F25\"-->\"%28x-4%2F5%29%5E2=121%2F25\"-->\"%28x-4%2F5%29%5E2=%2811%2F5%29%5E2\"
\n" ); document.write( "So either \"x-4%2F5=11%2F5\"-->\"x=4%2F5%2B11%2F5\"-->\"x=15%2F5\"-->\"highlight%28x=3%29\" ,
\n" ); document.write( "or \"x-4%2F5=-11%2F5\"-->\"x=4%2F5-11%2F5\"-->\"highlight%28x=-7%2F5%29\"
\n" ); document.write( "
\n" ); document.write( "THE QUADRATIC FORMULA says that the solutions to an equation of the form
\n" ); document.write( "\"ax%5E2%2Bbx%2Bc=0\" are given by
\n" ); document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"
\n" ); document.write( "That is easy enough if you
\n" ); document.write( "remember the formula,
\n" ); document.write( "have a calculator handy,
\n" ); document.write( "and have the knowledge and ability to use it without making mistakes.
\n" ); document.write( "In this case, we have \"5x%5E2-8x-21=0\" ,
\n" ); document.write( "so \"a=5\", \"b=-8\" and \"c=-21\" .
\n" ); document.write( "
\n" ); document.write( "The solutions are
\n" ); document.write( "\"x=%288%2B22%29%2F10=30%2F10=highlight%283%29\"
\n" ); document.write( "and \"x=%288-22%29%2F10=-14%2F10=highlight%28-7%2F5%29\" \n" ); document.write( "
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