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Mathematics Dictionary
Dr. K. G. Shih
Equation theory


  • Q01 | - Equation theory of quadratic equation
  • Q02 | - Equation theory of cubic equation
  • Q03 | - Equation theory of quartic equation
  • Q04 | - (pi^2)/6 = 1 + 1/(2^2) + 1/(3^2) + ....
  • Q05 | - (pi^2)/8 = 1 + 1/(3^2) + 1/(5^2) + ....

    • Q01. Equation theory of quadratic equation

    Equation : a*(x^2) + b*x + c = 0
    • Let roots be r and s
    • Then a*x^2 + b*x + c = 0
    • Then x^2 + (b/a)*x + c/a = (x - r)*(x - s) = 0
    • Then x^2 + (b/a)*x + c/a = x^2 - (r + s)*x + r*s = 0
    • Equation theory
      • Sum of roots = r + s = -b/a
      • Product of roots = r*s = c/a
      • 1/r + 1/s = -b/c
    Example : Roots of x^2 - 6*x + 8 = 0 are r and s. Find 1/r + 1/s
    • Method 1
      • x^2 - 6*x + 8 = (x - 2)*(x - 4) = 0
      • Hence r = 2 and s = 4
      • Hence 1/r + 1/s = 1/2 + 1/4 = 3/4
    • Method 2
      • 1/r + 1/s = -b/c = -(-6/8) = 3/4

    Go to Begin

    Q02. Equation theory of cubic equation

    Equation : a*(x^3) + b*(x^2) + c*x + d = 0
    • Let roots be r, s and t
    • Then a*x^2 + b*x + c = 0
    • Then x^3 + (b/a)*(x^2) + (c/a)*x + d/a = (x - r)*(x - s)*(x - t) = 0
    • Equation theory
      • Sum of roots = r + s + t = -b/a
      • Product of two roots = r*s + r*t + s*t = c/a
      • Product of three roots = -r*s*t
      • 1/r + 1/s + 1/t = -c/d
    Roots of x^3 - 6*(x^2) + 11*x - 6 = 0 are r, s and t. Find 1/r + 1/s + 1/t
    • Method 1
      • x^3 - 6*(x^2) + 11*x - 6 = (x - 1)*(x - 2)*(x - 3) = 0
      • Hence r = 1, s = 2 and t = 3
      • Hence 1/r + 1/s + 1/t = 1/1 + 1/2 +1/3 = 11/6
    • Method 2
      • 1/r + 1/s + 1/t = -c/d = -(-11/6) = 11/6

    Go to Begin

    Q03. Equation theory of quartic equation

    Equation : a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
    • Roots are r, s, t, u
    • Theory
      • r + s + t + u = -b/a
      • r*s + r*t + r*u + s*t + s*u + t*u = c/a
      • r*s*t + r*s*u + s*t*u = -d/a
      • r*s*t*u = e/a
      • 1/r + 1/s + 1/t + 1/u = -d/e
    Go to Begin

    Q04. (pi^2)/6 = 1 + 1/(2^2) + 1/(3^2) + ....

    Proof
    Go to Begin

    Q05. (pi^2)/8 = 1 + 1/(3^2) + 1/(5^2) + ....

    Proof
    Go to Begin
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