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1) Number Problem: Find two consecutive positive integers such that the sum of their squares is 85.
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\n" ); document.write( "The two consecutive numbers:
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\n" ); document.write( "Sum of their squares = 85:
\n" ); document.write( "x^2 + (x+1)^2 = 85
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\n" ); document.write( "x^2 + x^2 + 2x + 1 = 85; FOILed (x+1)(x+1)
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\n" ); document.write( "2x^2 + 2x + 1 - 85 = 0
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\n" ); document.write( "2x^2 + 2x - 84 = 0; a quadratic equation
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\n" ); document.write( "x^2 + x - 42; simplified, divided equation by 2
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\n" ); document.write( "(x + 7)(x - 6) = 0; easily factored
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\n" ); document.write( "x = +6 or x = -7; either solution will work
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\n" ); document.write( "numbers would be +6, +7 or -7, -6
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\n" ); document.write( "Check: 6^2 + 7^2 = 85 or -7^2 + -6^2 = + 85 also
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\n" ); document.write( "2) 3x^2-2x+1=0
\n" ); document.write( "This equation has no real roots. Look at the discriminant (b^2 - 4*a*c)
\n" ); document.write( "4 - 4*3*1; a negative value
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\n" ); document.write( "3) Construction: A garden area is 30 ft. long and 20 ft. wide. A path of uniform width is set around the edge. If the remaining garden area is 400 ft^2, what is the width of the path?
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\n" ); document.write( "Draw a diagram of this. Label the path width as x;
\n" ); document.write( "Label the rectangle 30 by 20, it will be apparent to you that the inner rectangle which is the garden, dimension will be (30-2x) by (20-2x)
\n" ); document.write( "That area is given as 400 sq/ft. So we have:
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\n" ); document.write( "(20-2x)*(30-2x) = 400
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\n" ); document.write( "600 - 100x + 4x^2 = 400
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\n" ); document.write( "Arrange as a quadratic:
\n" ); document.write( "4x^2 - 100x + 600 - 400 = 0
\n" ); document.write( "4x^2 - 100x + 200 = 0
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\n" ); document.write( "Simplify, divide equation by 4
\n" ); document.write( "x^2 - 25x + 50 = 0
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\n" ); document.write( "Use the quadratic formula to find x: a=1; b= -25; c=50
\n" ); document.write( "I assume you know how to do that:
\n" ); document.write( "I got approx values of x = 22.81 and x = 2.19, obviously it's the smaller value
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\n" ); document.write( "check this: 2x = 4.38, subtract that from 30 and 20 and get the dimension of the garden, find the area and see if it is, in fact, 400 sq/ft:
\n" ); document.write( "25.62 * 15.62 = 400.2 ~ 400 sq ft \n" ); document.write( "