Symmetric Functions of Roots of a Quadratic Equation

Instructor: Russell Frith
Let's explore the symmetric functions of the roots of a quadratic equation. In this lesson, we'll look at a list of essential symmetric functions, and examples will demonstrate how to convert a symmetric function into an equivalent function.

Functions of Roots of a Quadratic Equation

Let's say that we have x1 and x2, and they are the roots of the quadratic equation ax2 + bx + c = 0, (a <> 0). The expressions of the form x1 + x2, x12 + x22, x12 - x22, 1/x12 + 1/x22 and so forth are known as functions of the roots x1 and x2.

Finding the Roots of a Quadratic

So, how do we find the roots of a quadratic equation? We need to apply our knowledge of solving for any quadratic equation.


Symmetric Functions

Now, let's explore how to determine whether or not a quadratic's roots can form a symmetric function.

If the function using the roots of the quadratic f(x1,x2) doesn't change on interchanging x1 and x2, then the function, f, is symmetric. In other words, an expression in x1 and x2, which remains the same when x1 and x2 are interchanged, is called a symmetric function in x1 and x2.


For a quadratic equation ax2 + bx + c = 0, (a <> 0) with roots x1 and x2, we have both:

  • x1 + x2 = -b/a
  • x1*x2 = c/a

These two properties are used to formulate symmetric functions of the roots of a quadratic equation. When formulating those symmetric functions, we express them in terms of x1 + x2 and x1*x2.

Symmetric functions of quadratic roots include the following seven:

  • Formula (i): x12 + x22 = (x1 + x2)2 - 2x1 x2
  • Formula (ii): (x1 - x2)2 = (x1 + x2)2 - 4x1 x2
  • Formula (iii): x12 - x22 = (x1 + x2)(x1 - x2) = (x1 + x2) ((x1 + x2)2 - 4x1 x2)0.5
  • Formula (iv): x13 + x23 = (x1 + x2)3 - 3x1 x2 (x1 + x2)
  • Formula (v): x13 - x23 = (x1 - x2) (x12 + x1 x2 + x22)
  • Formula (vi): x14 + x24 = (x12 + x22)2 - 2 x12 x22
  • Formula (vii): x14 - x24 = (x1 + x2)(x1 - x2)(x12 + x22) = (x1 + x2)(x1 - x2) ((x12 + x22)2 - 2 x1 x2 )

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