document.write( "Question 108495: please help me on my homework.\r
\n" ); document.write( "\n" ); document.write( "1.the sum of the numbers is 15. the sum of the squares of the numbers is 113. find the numbers.\r
\n" ); document.write( "\n" ); document.write( "2. one number is 6 larger than the other number. the square of the larger number is 132 greater than the square of the smaller number. what are the numbers?\r
\n" ); document.write( "\n" ); document.write( "3. 5 years hence, a boy's age will be 4 times the square of his age 9 years ago. how old is this boy?\r
\n" ); document.write( "\n" ); document.write( "4. the perimeter of a square is numerically 4 more than its area. find the length of its side.\r
\n" ); document.write( "\n" ); document.write( "5. the width of a rectangle is 9 cm shorter than its diagonal and 7 cm shorter than its length. how long is the diagonal? \n" ); document.write( "
Algebra.Com's Answer #79149 by ankor@dixie-net.com(22740)\"\" \"About 
You can put this solution on YOUR website!
1.the sum of the numbers is 15. the sum of the squares of the numbers is 113. find the numbers.
\n" ); document.write( ":
\n" ); document.write( "Write the equation for the statements,
\n" ); document.write( "\"The sum of the numbers is 15\"
\n" ); document.write( "x + y = 15
\n" ); document.write( "or
\n" ); document.write( "y = (15 - x);we can use this for substitution
\n" ); document.write( ":
\n" ); document.write( "\"The sum of the squares is 113\"
\n" ); document.write( "x^2 + y^2 = 113
\n" ); document.write( ":
\n" ); document.write( "Substitute (15-x) for y in the above equation
\n" ); document.write( "x^2 + (15-x)^2 = 113
\n" ); document.write( ":
\n" ); document.write( "FOIL (15-x)((15-x)
\n" ); document.write( "x^2 + 225 - 30x + x^2 = 113
\n" ); document.write( "Arrange as a quadratic equation:
\n" ); document.write( "x^2 + x^2 - 30x + 225 - 113 = 0
\n" ); document.write( "2x^2 - 30x + 112 = 0
\n" ); document.write( ":
\n" ); document.write( "Simplify, divide by 2
\n" ); document.write( "x^2 - 15x + 56 = 0
\n" ); document.write( "This factors to:
\n" ); document.write( "(x - 8)(x - 7) = 0
\n" ); document.write( ":
\n" ); document.write( "x = 8 or x = 7
\n" ); document.write( "Then
\n" ); document.write( "y = 7 or y = 8
\n" ); document.write( ":
\n" ); document.write( "Check solution: 8^2 + 7^2 = 113
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( "2. one number is 6 larger than the other number. the square of the larger number is 132 greater than the square of the smaller number. what are the numbers?
\n" ); document.write( ":
\n" ); document.write( "Let x = \"one number\"
\n" ); document.write( "Then
\n" ); document.write( "(x-6) = \"the other number
\n" ); document.write( ":
\n" ); document.write( "Write an equation for what it says:
\n" ); document.write( "\"the square of the larger number is 132 greater than the square of the smaller number.\"
\n" ); document.write( "x^2 = (x-6)^2 + 132
\n" ); document.write( "x^2 = x^2 - 12x + 36 + 132
\n" ); document.write( "Arrange the x's on the left
\n" ); document.write( "x^2 - x^2 + 12x = 168
\n" ); document.write( "12x = 168
\n" ); document.write( "x = 168/12
\n" ); document.write( "x = 14
\n" ); document.write( "You can figure out what the smaller number is; check your solution
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( "3. 5 years hence, a boy's age will be 4 times the square of his age 9 years ago. how old is this boy?
\n" ); document.write( ":
\n" ); document.write( "Let x = the boys age now
\n" ); document.write( ":
\n" ); document.write( "\"5 years hence\" he will be:
\n" ); document.write( "(x + 5)
\n" ); document.write( ":
\n" ); document.write( "\"9 year ago\" he was:
\n" ); document.write( "(x - 9)
\n" ); document.write( ";
\n" ); document.write( "Write an equation for the statement:
\n" ); document.write( "(x + 5) = 4(x-9)^2
\n" ); document.write( "x + 5 = 4(x^2 - 18x + 81)
\n" ); document.write( "x + 5 = 4x^2 - 72x + 324
\n" ); document.write( "Arrange as a quadratic:
\n" ); document.write( "4x^2 - 72x - x + 324 - 5 = 0
\n" ); document.write( "4x^2 - 73x + 319 = 0
\n" ); document.write( "you probably need the quadratic formula to find x, but it does factor:
\n" ); document.write( "(4x - 29)(x - 11) = 0
\n" ); document.write( ":
\n" ); document.write( "x = 11, it's the integer solution we want here.
\n" ); document.write( ":
\n" ); document.write( "Check: 16 = 4(11-9)
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( "4. the perimeter of a square is numerically 4 more than its area. find the length of its side.
\n" ); document.write( ":
\n" ); document.write( "Let x = the length of the side
\n" ); document.write( "Then
\n" ); document.write( "4x = the perimeter
\n" ); document.write( "and
\n" ); document.write( "x^2 = the area
\n" ); document.write( ":
\n" ); document.write( "The statement says:
\n" ); document.write( "4x = x^2 + 4
\n" ); document.write( "Arrange as a quadratic equation:
\n" ); document.write( "x^2 - 4x + 4 = 0
\n" ); document.write( "Factors to:
\n" ); document.write( "(x-2)(x-2)
\n" ); document.write( "obviously x = +2
\n" ); document.write( ":
\n" ); document.write( "Check: 4(2) = 2^2 + 4
\n" ); document.write( ":
\n" ); document.write( ":
\n" ); document.write( "5. the width of a rectangle is 9 cm shorter than its diagonal and 7 cm shorter than its length. how long is the diagonal?
\n" ); document.write( ":
\n" ); document.write( "Let x = the diagonal
\n" ); document.write( "Then
\n" ); document.write( "(x-9) = width
\n" ); document.write( ":
\n" ); document.write( "It says the length is 7 cm longer than the width. which is (x-9), therefore
\n" ); document.write( "(x-9) + 7 = (x-2) is the length
\n" ); document.write( ":
\n" ); document.write( "Using pythag we have:
\n" ); document.write( "Length^2 + width^2 = diagonal^2
\n" ); document.write( "(x-2)^2 + (x-9)^2 = x^2
\n" ); document.write( ":
\n" ); document.write( "See if you can do this yourself by what was shown above, only one of the solutions will make sense. \r
\n" ); document.write( "\n" ); document.write( "If you have any difficulty you can email me. \n" ); document.write( "
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