The ex-radii of a triangle are 10 1 2 , 12 and 14 . If the sides of the triangle are the roots of the cubic x^3-p x^2+q x-r=0 , where p, q, r are integers, find the integer nearest to p+q+r .

Step-by-step solution

r₁= S-a = 21 2 , r₂= S-b =12 and r₃= S-c =14 On solving above equations, we get a=13, b=14, c=15 Let f(x)=x^3-p x^2+q x-r=(x-13)(x-14)(x-15)f(-1)=-1-p-q-r=(-14)(-15)(-16) p+q+r=14 15 16-1 p+q+r = 3364 58