If one of the side ribs, which equals x² is hypotenuse, then the other rib length is: b 2 = ( x 2 ) 2 − ( 6 x ) 2 = x 4 − 36 x 2 . b^2 = (x^2)^2 - (6x)^2 = x^4 - 36x^2. b 2 = ( x 2 ) 2 − ( 6 x ) 2 = x 4 − 36 x 2 .
b = x 4 − 36 x 2 b = \sqrt{x^4 - 36x^2} b = x 4 − 36 x 2
The perimeter of this triangle is:
P = x 2 + 6 x + x 4 − 36 x 2 = 80 , P = x^2 + 6x + \sqrt{x^4 - 36x^2} = 80, P = x 2 + 6 x + x 4 − 36 x 2 = 80 ,
x 4 − 36 x 2 = 80 − x 2 − 6 x , \sqrt{x^4 - 36x^2} = 80 - x^2 - 6x, x 4 − 36 x 2 = 80 − x 2 − 6 x ,
x 4 − 36 x 2 = 6 , 400 − 80 x 2 − 480 x − 80 x 2 + x 4 + 6 x 3 − 480 x + 6 x 3 + 36 x 2 , x^4 - 36x^2 = 6,400 - 80x^2 - 480x - 80x^2 + x^4 + 6x^3 - 480x + 6x^3 + 36x^2, x 4 − 36 x 2 = 6 , 400 − 80 x 2 − 480 x − 80 x 2 + x 4 + 6 x 3 − 480 x + 6 x 3 + 36 x 2 ,
12 x 3 − 88 x 2 − 960 x + 6 , 400 = 0 , 12x^3 - 88x^2 - 960x + 6,400 = 0, 12 x 3 − 88 x 2 − 960 x + 6 , 400 = 0 ,
3 x 3 − 22 x 2 − 240 x + 1 , 600 = 0 , 3x^3 - 22x^2 - 240x + 1,600 = 0, 3 x 3 − 22 x 2 − 240 x + 1 , 600 = 0 ,
( 3 x 3 − 30 x 2 ) + ( 8 x 2 − 240 x + 1 , 600 ) = 0 , (3x^3 - 30x^2) + (8x^2- 240x + 1,600) = 0, ( 3 x 3 − 30 x 2 ) + ( 8 x 2 − 240 x + 1 , 600 ) = 0 ,
3 x 2 ( x − 10 ) + 8 ( x 2 − 30 x + 200 ) = 0 , 3x^2(x - 10) + 8(x^2 - 30x + 200) = 0, 3 x 2 ( x − 10 ) + 8 ( x 2 − 30 x + 200 ) = 0 ,
3 x 2 ( x − 10 ) + 8 ( x − 10 ) ( x − 20 ) = 0 , 3x^2(x - 10) + 8(x - 10)(x - 20) = 0, 3 x 2 ( x − 10 ) + 8 ( x − 10 ) ( x − 20 ) = 0 ,
( x − 10 ) ( 3 x 2 + 8 x − 160 ) = 0 , (x - 10)(3x^2 + 8x - 160) = 0, ( x − 10 ) ( 3 x 2 + 8 x − 160 ) = 0 ,
So, x1 = 10 is one of the roots of this equation.
D = 64 + 1920 = 1984.
x 2 = − 8 + 1984 6 = 6.09 , x2 = \frac{-8 + \sqrt{1984}}{6} = 6.09, x 2 = 6 − 8 + 1984 = 6.09 ,
x 3 = − 8 − 1984 6 = − 8.76 , x3 = \frac{-8 - \sqrt{1984}}{6} = -8.76, x 3 = 6 − 8 − 1984 = − 8.76 , which is not possible in our case.
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