Sides of the triangle

Calculate triangle sides where its area is S = 84 cm2 and a = x, b = x + 1, xc = x + 2

Result

a =  13 cm
b =  14 cm
c =  15 cm

Solution:

S=s(s−a)(s−b)(s−c)=84 s=(a+b+c)/2=(3x+3)/2=1.5x+1.5 (1.5x+1.5)(1.5x+1.5−x)(1.5x+1.5−x−1)(1.5x+1.5−x−2)=842 (1.5x+1.5)(0.5x+1.5)(0.5x+0.5)(0.5x−0.5)=7056  0.1875x4+0.75x3+0.375x2−0.75x−7056.56=0  x=13 a=x=13 cm S = \sqrt{s(s-a)(s-b)(s-c)} = 84 \ \\ s = (a+b+c)/2 = (3x+3)/2 = 1.5x +1.5 \ \\ (1.5x +1.5)(1.5x +1.5-x)(1.5x +1.5-x-1)(1.5x +1.5-x-2) = 84^2 \ \\ (1.5x +1.5)(0.5x+1.5)(0.5x+0.5)(0.5x-0.5)= 7056 \ \\ \ \\ 0.1875 x^4+0.75 x^3+0.375 x^2-0.75 x-7056.56 = 0 \ \\ \ \\ x = 13 \ \\ a = x = 13 \ \text{cm} \ \\
b=13+1=14 cmb=13+1=14 \ \text{cm}
c=13+2=15 cmc=13+2=15 \ \text{cm}

Equation is non-linear.
Equation is not quadratic.
Equation is not cubic.

The real roots of the equation:

x1 = -14.9999987727
x2 = 12.9999987727

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