f (x) = ax

The graph of such a function is given below. For genuine cubics with non-zero coefficient a, dividing through a and deleting the primes from b' = b/a, c' = c/a, d' = d/a leads to

x

x

and results from the previous one by eliminating the quadratic term. This is done by translating the origin through the change of variable

x = x' - (b/3).

Introducing this into the equation of the previous section and doing the calculation gives p = (3c-b

q = (9bc - 2b

The graph of the function f(x) = x

D = (p/3)

Consequently the equation has three real roots only when

(p/3)

Substitution of the expressions for (p,q) results to the equation

(27*d

An interesting application of this inequality occurs in the problem of constructing a triangle from the so-called

In this case the side-lengths of the triangle appear as the three real solutions of a cubic equation (see section-4 of GIO_Cnstruction.html ). The coefficients of this cubic are

b = -2s,

c = (s

d = -4Rrs.

Substitution of these into the above inequality leads to the necessary condition for the three {s,r,R} in order to determine a triangle:

(64*r*R

The graph of the function above depends on the position of the red point. The coordinates of this point serve to define the pair (p,q).

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