document.write( "Question 874959: pleae answe these qustion
\n" ); document.write( "1.find two numbers whose sum is 13 and whose product is 40 \r
\n" ); document.write( "\n" ); document.write( "2.the area of a rectangle is 94sq. units and its perimeter is 38units.
\n" ); document.write( "find the length and width of the rectangle.
\n" ); document.write( "thanks \n" ); document.write( "
Algebra.Com's Answer #527865 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
1. You can set up a quadratic equation and solve it, and that may be what is expected, but it is not the most efficient way.
\n" ); document.write( "THE FIFTH GRADER WAY:
\n" ); document.write( "The most effective way would be to find pairs of factors whose product is \"40\"
\n" ); document.write( "and see if the sum of one of those pairs is \"13\" :
\n" ); document.write( "\"40=40%2A1\" and \"40%2B1=41%3C%3E13\" so it is not \"40+and+%7B%7B%7B1\"
\n" ); document.write( "\"40=20%2A2\" and \"20%2B2=22%3C%3E13\" so it is not \"20+and+%7B%7B%7B2\"
\n" ); document.write( "\"40=10%2A4\" and \"10%2B4=14%3C%3E13\" so it is not \"10+and+%7B%7B%7B4\"
\n" ); document.write( "\"40=8%2A5\" and \"8%2B5=13\" so the answer is \"highlight%288%29\" and \"highlight%2813%29\" .
\n" ); document.write( "I did not even need to mention \"quadratic equations\" or know what the term means.
\n" ); document.write( "INVOKING QUADRATIC EQUATIONS:
\n" ); document.write( "\"x\"= one of the numbers
\n" ); document.write( "\"y\"= the other number
\n" ); document.write( "Since we know they add to \"13\" ,
\n" ); document.write( "\"x%2By=13\"--->\"y=13-x\"
\n" ); document.write( "So the product of the two numbers is
\n" ); document.write( "\"40=x%2813-x%29\"
\n" ); document.write( "\"40=13x-x%5E2\"
\n" ); document.write( "\"highlight%28x%5E2-13x%2B40=0%29\"
\n" ); document.write( "I see 3 ways to solve that quadratic equation.
\n" ); document.write( "1) I would factor it into \"%28x-8%29%28x-5%29=0\"
\n" ); document.write( "and then make each factor equal to zero.
\n" ); document.write( "Factoring really means doing the work the fifth grader would do,
\n" ); document.write( "and I believe that is the most efficient way to solve this particular quadratic equation.
\n" ); document.write( "\"x-5=0%29\"--->\"highlight%28x=5%29\"-->\"y=13-x=13-5\"--->\"highlight%28y=8%29\"
\n" ); document.write( "\"x-8=0%29\"--->\"highlight%28x=8%29\"-->\"y=13-x=13-8\"--->\"highlight%28y=5%29\"
\n" ); document.write( "Either solution tells us that the two numbers are \"highlight%288%29\" and \"highlight%285%29\" .
\n" ); document.write( "2) we could use the quadratic formula that says that the solutions to
\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( "For \"x%5E2-13x%2B40=0\" , \"a=1\" , \"b=-13\" , and \"c=40\" , so
\n" ); document.write( "
\n" ); document.write( "That gives us the answers
\n" ); document.write( "\"x=%2813%2B3%29%2F2\"--->\"x=16%2F2\"--->\"highlight%28x=8%29\" and
\n" ); document.write( "\"x=%2813-3%29%2F2\"--->\"x=10%2F2\"--->\"highlight%28x=5%29\" as above.
\n" ); document.write( "3) The third option is completing the square:
\n" ); document.write( "\"x%5E2-13x%2B40=0\"
\n" ); document.write( "\"x%5E2-13x=-40\"
\n" ); document.write( "\"x%5E2-13x%2B%2813%2F2%29%5E2=-40%2B%2813%2F2%29%5E2\"
\n" ); document.write( "\"%28x-13%2F2%29%5E2=-40%2B169%2F4\"
\n" ); document.write( "\"%28x-13%2F2%29%5E2=-160%2F4%2B169%2F4\"
\n" ); document.write( "\"%28x-13%2F2%29%5E2=9%2F4\"
\n" ); document.write( "\"%28x-13%2F2%29%5E2=%283%2F2%29%5E2\"
\n" ); document.write( "So either \"x-13%2F2=3%2F2\"--->\"x=13%2F2%2B3%2F2\"--->\"x=16%2F2\"--->\"highlight%28x=8%29\" or
\n" ); document.write( "\"x-13%2F2=-3%2F2\"--->\"x=13%2F2-3%2F2\"--->\"x=10%2F2\"--->\"highlight%28x=5%29\" .
\n" ); document.write( "Any way you solve the quadratic equation, the two numbers are \"highlight%288%29\" and \"highlight%285%29\" .
\n" ); document.write( "
\n" ); document.write( "2. You have to find two numbers that are the measurements of \"length\" and \"width\" for that rectangle.
\n" ); document.write( "\"area=length%2Awidth\"
\n" ); document.write( "so \"length%2Awidth=94\"
\n" ); document.write( "\"perimeter=2%28length%2Bwidth%29\"
\n" ); document.write( "so \"2%28length%2Bwidth%29=38\"
\n" ); document.write( "\"length%2Bwidth=38%2F2\"
\n" ); document.write( "\"length%2Bwidth=19\"
\n" ); document.write( "THE FIFTH GRADER WAY/WITHOUT MENTIONING QUADRATIC EQUATIONS:
\n" ); document.write( "Could it be a square?
\n" ); document.write( "If it was a square with a side length of \"x\" units,
\n" ); document.write( "The perimeter would be \"4x=38\"--->\"x=38%2F4\"-->\"x=9.5\" units
\n" ); document.write( "The area of that square would be \"x%2Ax=9.5%2A9.5=90.25\" square units .
\n" ); document.write( "A square with a perimeter of \"38\" units has an area of \"90.25\" square units, and we are asked to find a larger rectangle with the same perimeter and an area of \"94\" square units .
\n" ); document.write( "There is something wrong with this problem.
\n" ); document.write( "In the real world there is no such a rectangle.
\n" ); document.write( "Maybe there was a typo, or maybe it is a trick question,
\n" ); document.write( "because we know, from all those fencing-a-square-pen problems, that the largest rectangle we can make with a given perimeter is a square.
\n" ); document.write( "There are no real length and width measurements for that rectangle.
\n" ); document.write( "Could it be that the measurements are imaginary numbers?
\n" ); document.write( "The fifth grader goes to the computer to do a search on \"imaginary numbers.\"
\n" ); document.write( "
\n" ); document.write( "THE QUADRATIC WAY:
\n" ); document.write( "So the sum of those two measures is \"19\" and the product is \"94\" .
\n" ); document.write( "This problem is similar to the one above, but factoring does not always work.
\n" ); document.write( "Sometimes factoring is complicated,
\n" ); document.write( "Sometimes the solutions are irrational numbers, so you would have to write a quadratic equation and try to solve it by using the quadratic formula or completing the square.
\n" ); document.write( "Sometimes there are no real solutions.
\n" ); document.write( "In all those cases, the quadratic formula (or completing the square) would tell us the answer.
\n" ); document.write( "If one of those measures is \"x\" , the other will be \"19-x\"
\n" ); document.write( "and \"94=x%2819-x%29\"--> \"94=19x-x%5E2\"-->\"highlight%28x%5E2-19x%2B94=0%29\"
\n" ); document.write( "To apply the quadratic formula,
\n" ); document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\" ,
\n" ); document.write( "for this equation \"a=1\" , \"b=-19\" , and \"c=94\" .
\n" ); document.write( "Applying the quadratic formula,
\n" ); document.write( "\"x+=+%28-%28-19%29+%2B-+sqrt%28%28-19%29%5E2-4%2A1%2A94+%29%29%2F%282%2A1%29+\"
\n" ); document.write( "\"x+=+%2819+%2B-+sqrt%28361-376%29%29%2F2\"
\n" ); document.write( "\"x+=+%2819+%2B-+sqrt%28-15%29%29%2F2\"
\n" ); document.write( "We have no real solutions, because \"sqrt%28-15%29\" is no real number,
\n" ); document.write( "so there are no real solutions, but we could ask that fifth grader about imaginary ones. \n" ); document.write( "
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