Answer to Question #174396 in Algebra for Prosper

Question

If q is a double root of the equation

x² + ax + b=0. Prove that

a=-3q², b= 2q³ and hence

4a³ + 27b²=0

Answer 1

The maths formulae can be proved by using the completing the square method with the correspondent variables

• : that is a and b

The discriminant of a quadratic equation

        ax² + bx + c = 0

is

        Δ = b² – 4ac.

If the discriminant Δ is zero, the equation has a double root (i.e. there is a unique x that makes the equation zero, and it counts twice as a root). If the discriminant is not zero, there are two distinct roots.

Cubic equations also have a discriminant. For a cubic equation

        ax³ + bx² + cx + d = 0

the discriminant is given by

        Δ = 18abcd – 4b³d + b²c² – 4ac³ – 27a²d².

If Δ = 0, the equation has a multiple root. Otherwise, it has three distinct roots.

A change of variable can reduce the general cubic equation to a so-called “depressed” cubic equation of the form

        x³ + px + q = 0

in which case the discriminant simplifies to

        Δ = – 4p³ – 27q².

Here are a couple of interesting connections. The idea of reducing a cubic equation to a depressed cubic goes back to Cardano (1501–1576). What’s called a depressed cubic in this context is known as the Weierstrass (1815–1897) form in the context of elliptic curves. That is, an elliptic curve of the form

        y² = x³ + ax + b

is said to be in Weierstrass form. In other words, an elliptic curve is in Weierstrass form if the right-hand side is a depressed cubic.

Furthermore, an elliptic curve is required to be non-singular, which means it must satisfy

        4a³ + 27b² ≠ 0.

In other words, the discriminant of the right-hand side is non-zero. In the context of elliptic curves, the discriminant is defined to be

        Δ = -16(4a³ + 27b²)

The Discriminant of a Cubic

The discriminant of a quadratic equation

        ax² + bx + c = 0

is

        Δ = b² – 4ac.

If the discriminant Δ is zero, the equation has a double root (i.e. there is a unique x that makes the equation zero, and it counts twice as a root). If the discriminant is not zero, there are two distinct roots.

Cubic equations also have a discriminant. For a cubic equation

        ax³ + bx² + cx + d = 0

the discriminant is given by

        Δ = 18abcd – 4b³d + b²c² – 4ac³ – 27a²d².

If Δ = 0, the equation has a multiple root. Otherwise, it has three distinct roots.

A change of variable can reduce the general cubic equation to a so-called “depressed” cubic equation of the form

        x³ + px + q = 0

in which case the discriminant simplifies to

        Δ = – 4p³ – 27q².

Here are a couple of interesting connections. The idea of reducing a cubic equation to a depressed cubic goes back to Cardano (1501–1576). What’s called a depressed cubic in this context is known as the Weierstrass (1815–1897) form in the context of elliptic curves. That is, an elliptic curve of the form

        y² = x³ + ax + b

is said to be in Weierstrass form. In other words, an elliptic curve is in Weierstrass form if the right-hand side is a depressed cubic.

Furthermore, an elliptic curve is required to be non-singular, which means it must satisfy

        4a³ + 27b² ≠ 0.

In other words, the discriminant of the right-hand side is non-zero. In the context of elliptic curves, the discriminant is defined to be

        Δ = -16(4a³ + 27b²)

which is the same as the discriminant above, except for a factor of 16 that simplifies some calculations with elliptic curves.