cubic equation 113

2

(u3+ v3) + (3uv + p)(u+ v) + q= 0

This equation will be satisfied if we can choose uand v

so that u3+ v3= –qand 3uv + p= 0. This yields a pair

of equations for u3and v3:

Solving for v3in the first equation and substituting the

result into the second shows that u3must satisfy the

quadratic equation:

and using the quadratic formula, this gives u3, and con-

sequently v3= –u3– q, to be the two numbers:

We must now take the cube root of these quantities.

Note first that any number Mhas three cube roots: one

real, denoted 3

√

–

M, and two imaginary, w×3

√

–

M, and

w2×3

√

–

M, where . Set:

and

One can now check that the three quantities u1+ v1, u2

+ v2, and u3+ v3represent the three solutions to the

reduced cubic y3+ py + q= 0. They constitute Car-

dano’s formula.

The quantity under the square root sign:

is called the discriminant of the cubic, and it determines

the nature of the solutions:

If ∆> 0, then the equation has one real root

and two complex roots.

If ∆= 0, then the equation has three real roots,

at least two of which are equal.

If ∆< 0, then the equation has three distinct

real roots.

In the third case, one is required to take combinations

of cube roots of complex numbers to yield, surpris-

ingly, purely real answers. For example, Cardano’s

method applied to the equation x3= 15x+ 4 yields as

one solution the quantity:

It is not immediate that this number is x= 4.

This confusing phenomenon of using complex

quantities to produce real results was first explored by

Italian mathematician R

AFAEL

B

OMBELLI

(1526–72).

French mathematician F

RANÇOISE

V

IÈTE

(1540–1603)

used trigonometric formulae as an alternative approach

to identifying the three distinct real roots that appear in

this puzzling scenario.

Another Method

French mathematician Viète also developed the follow-

ing simpler approach to solving cubic equations. This

method was published posthumously in 1615.