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Mathematics

Symmetric Properties of Roots – Quadratic equation

Certain relations involving α and β can also be determined from α+β andαβ even when we do not know α and β distinctly. Such relations are usually said to be symmetric. They are symmetric in the sense that if α and α are interchanged, either the relation remains the same or is multiplied by -1

Example – If  α ≠ β, determine whether or not each of the following is symmetric

(a) α+β (b) αβ (c) α2 + β2 (d) α2 – β2 (e) 3α +2β (f) α3 + β3

Solution

  • α+β = β +α , therefore α+β is symmetric
  • αβ =βα, therefore αβ is symmetric
  • α2 + β2 = β2 + α2, therefore α2 + β2 is symmetric
  • α2 – β2 = – (β2 – α2), thereforeα2 – β2 is symmetric
  • 3α +2β ≠ 3β + 2α since α ≠ β, therefore 3α +2β is not symmetric
  • α3 + β3 = β3 + α3,  therefore α3 + β3 is symmetric

EVALUATION (USE THE BOX AT THE BOTTOM TO POST YOUR ANSWER FOR DISCUSSION AND APPRAISAL)

  1. Find the quadratic equation whose roots are ½ and 5
  2. Find the sum & product of roots of the equation 3x2 – 5x – 2 = 0

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