Solving a Quadratic Equation¶

For

$$ ax^2+bx+c=0,\qquad a\ne0, $$

the two roots are

$$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. $$

Rather than entering roots separately for each example, the Egison program reads the coefficients of a polynomial, constructs its discriminant, and applies one typed symbolic solver.

Coefficients and the discriminant¶

Egison's coefficients function returns coefficients in ascending degree order. Thus the pattern $[a_0,a_1,a_2]$ recognizes $a_2x^2+a_1x+a_0$. The helper keeps the cleared-denominator discriminant $b^2-4ac$ intact and explicitly groups the denominator as $2a$.

declare symbol x, a, b, c: MathValue

def solveQuadraticCoefficientsDemo
  (a : MathValue)
  (b : MathValue)
  (c : MathValue)
  : (MathValue, MathValue) :=
  let discriminant := b ^ 2 - 4 * a * c
   in ( ((- b) + sqrt discriminant) / (2 * a)
      , ((- b) - sqrt discriminant) / (2 * a) )

def solveQuadraticDemo
  (f : MathValue)
  (x : MathValue)
  : (MathValue, MathValue) :=
  match coefficients f x as list mathValue with
    | [$a_0, $a_1, $a_2] ->
      solveQuadraticCoefficientsDemo a_2 a_1 a_0

A cyclotomic example¶

The polynomial $x^2+x+1$ has discriminant $-3$. Its roots are the two primitive cube roots of unity.

solveQuadraticDemo (x ^ 2 + x + 1) x

$(\frac{1}{2} \sqrt{3} i + \frac{-1}{2}, \frac{-1}{2} \sqrt{3} i + \frac{-1}{2})$

The symbolic formula¶

Leaving $a$, $b$, and $c$ symbolic exposes the usual discriminant without any special formatting code.

solveQuadraticDemo (a * x ^ 2 + b * x + c) x

$(\frac{-1}{2} a^{-1} b + \frac{1}{2} \sqrt{b^{2} - 4 a c} a^{-1}, \frac{-1}{2} a^{-1} b + \frac{-1}{2} \sqrt{b^{2} - 4 a c} a^{-1})$

A useful rescaling¶

Writing the middle coefficient as $2b$ gives the equivalent compact form

$$ x=\frac{-b\pm\sqrt{b^2-ac}}{a}. $$

solveQuadraticDemo (a * x ^ 2 + 2 * b * x + c) x

$(-a^{-1} b + \sqrt{b^{2} - a c} a^{-1}, -a^{-1} b - \sqrt{b^{2} - a c} a^{-1})$

Takeaway¶

Pattern matching and the typed helper separate the calculation into two transparent stages: read the polynomial coefficients, then form the discriminant and both signs of its square root. The displayed answers are produced from each input polynomial rather than inserted as precomputed outputs.