document.write( "Question 626824: The rectangular floor of a
\n" ); document.write( "shed has a length 4 feet longer than its width, and its
\n" ); document.write( "area is 140 square feet. Let x be the width of the floor.
\n" ); document.write( "(a) Write a quadratic equation whose solution gives
\n" ); document.write( "the width of the floor.
\n" ); document.write( "(b) Solve this equation.\r
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Algebra.Com's Answer #394484 by KMST(5328)\"\" \"About 
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Let \"x\" be the width of the floor (in feet).
\n" ); document.write( "The length is 4 feet longer than the width, so it is
\n" ); document.write( "\"x%2B4\" (in feet).
\n" ); document.write( "(a) The area in square feet can be calculated by multiplying length (in feet) times width (in feet), and it equals 140 square feet, so
\n" ); document.write( "\"%28x%2B4%29%2Ax=140\"
\n" ); document.write( "That equation can be transformed into a more traditional form:
\n" ); document.write( "\"%28x%2B4%29%2Ax=140\" --> \"x%5E2%2B4x=140\" --> \"highlight%28x%5E2%2B4x-140=0%29\"
\n" ); document.write( "
\n" ); document.write( "(b) The equation can be solved by factoring, or by completing the square, or by using the quadratic formula. I'll show you all 3 algebra ways of solving a quadratic equation, plus the way a fifth grader could solve the problem.
\n" ); document.write( "
\n" ); document.write( "Someone who does not know algebra would not be able to write the equation, but could look for two numbers that multiply to give 140 and could be the width and length in feet. Possible products are:
\n" ); document.write( "\"1%2A140=140\"
\n" ); document.write( "\"2%2A70=140\"
\n" ); document.write( "\"4%2A35=140\"
\n" ); document.write( "\"5%2A28=140\"
\n" ); document.write( "\"7%2A20=140\"
\n" ); document.write( "\"10%2A14=140\"
\n" ); document.write( "The last one of those products represents the width (10 feet) times the length (14 feet, 4 more than the width) of the rectangle.
\n" ); document.write( "
\n" ); document.write( "FACTORING (when it works, the easiest way):
\n" ); document.write( "The person who knows algebra (and became good at factoring polynomials) will do something similar, looking for integers that multiply to give the \"-140\" in the equation, and that will add up to the \"4\" in the equation
\n" ); document.write( "\"x%5E2%2B4x-140=0\"
\n" ); document.write( "The integers, \"-10\" and \"14\", are the numbers added to x in the factors of the quadratic polynomial:
\n" ); document.write( "\"x%5E2%2B4x-140=%28x-10%29%28x%2B14%29\"
\n" ); document.write( "The factored equation is
\n" ); document.write( "\"%28x-10%29%28x%2B14%29=0\" with solutions that make \"x-10=0\" or \"x%2B14=0\":
\n" ); document.write( "\"highlight%28x=10%29\" and \"x=-14\"
\n" ); document.write( "The only solution to the problem is
\n" ); document.write( "\"x-10=0\" --> \"highlight%28x=10%29\" (The width is 10 feet),
\n" ); document.write( "because the width in feet cannot be a negative number.
\n" ); document.write( "While \"x=-14\" is a solution to \"x%2B14=0\" and to the quadratic equation, it is not a solution to part (b) of the problem.
\n" ); document.write( "
\n" ); document.write( "COMPLETING THE SQUARE:
\n" ); document.write( "\"x%5E2%2B4x-140=0\" --> \"x%5E2%2B4x=140\"
\n" ); document.write( "Then, adding 4 to both sides of the equal sign,
\n" ); document.write( "we get \"x%5E2%2B4x%2B4=%28x%2B2%29%5E2\" on the left side.
\n" ); document.write( "\"x%5E2%2B4x=140\" --> \"x%5E2%2B4x%2B4=144\" --> \"%28x%2B2%29%5E2=144\"
\n" ); document.write( "As \"144=12%5E2\" or \"sqrt%28144%29=12\",
\n" ); document.write( "\"%28x%2B2%29%5E2=144\" --> \"x%2B2=12\" or \"x%2B2=-12\"
\n" ); document.write( "\"x%2B2=12\" --> \"x%2B2-2=12-2\"--> \"highlight%28x=10%29\"
\n" ); document.write( "\"x%2B2=-12\" --> \"x=-14\", which is not a solution to the problem, because the width cannot be -14 feet.
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\n" ); document.write( "USING THE QUADRATIC FORMULA:
\n" ); document.write( "The solutions to an equation of the form \"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( "In the case of equation \"x%5E2%2B4x-140=0\",
\n" ); document.write( "\"a=1\", \"b=4\", and \"c=-140\"
\n" ); document.write( "So \"x+=+%28-4+%2B-+sqrt%28+4%5E2-4%2A1%2A%28-140%29+%29%29%2F%282%2A1%29+\"
\n" ); document.write( "\"x+=+%28-4+%2B-+sqrt%2816%2B560%29%29%2F2\"
\n" ); document.write( "\"x+=+%28-4+%2B-+sqrt%28576%29%29%2F2\"
\n" ); document.write( "\"x+=+%28-4+%2B-+24%29%2F2\"
\n" ); document.write( "\"x+=+%28-4%2B24%29%2F2\" --> \"x=20%2F2\" --> \"highlight%28x=10%29\"
\n" ); document.write( "\"x+=+%28-4-24%29%2F2\" --> \"x=-28%2F2\" --> \"x=-14\", which is not a solution to the problem, because the width cannot be -14 feet. \n" ); document.write( "
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