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 + α^3,  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