Relations Between Roots Solution3





The page relations between roots solution3 is containing solution of some practice questions from the worksheet relationship between roots and coefficients.

(3) If α and β are the roots of the equation 3 x² - 5 x + 2 =0 then find the values of

(i) (α/β) + (β/α)

(ii) α - β

(iii) (α²/β) + (β²/α)

Solution:

3 x² - 5 x + 2 = 0

To get the values a,b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

 a = 3  b = - 5 and c = 2

Sum of roots α + β = -b/a

                           = - (-5)/3

                           = 5/3

Product of roots α β = c/a 

                            = 2/3

(i) (α/β) + (β/α) = (α² + β²)/αβ

            α² + β² = (α + β)² - 2 αβ

                       = (5/3)² - 2 (2/3)

                       = 25/9 - 4/3

                       = (25/9) - [4(3)/9]

                       = (25 - 12)/9

                       = 13/9

  (α/β) + (β/α) = (α² + β²)/αβ

                     = (13/9)/(2/3)

                     = 13/9 x 3/2

                     = 13/6

 (ii) α - β

   α - β = √(α + β)² - 4 α β

           = √(5/3)² - 4 (2/3)

           = √(25/9) - (8/3)

           = √(25/9) - (24/9)

           = √(25 - 24/9)

           = √1/9

           = ± 1/3

(iii) (α²/β) + (β²/α) = (α³ + β³)/αβ

   α³ + β³ = (α + β)³ - 3 αβ (α + β)

              = (5/3)³ - 3 (2/3) (5/3)

              = 125/27 - 10/9

              = (125 -30)/27

              = 95/27

(α²/β) + (β²/α) = (α³ + β³)/αβ

                     = (95/27)/(2/3)

                     = (95/27) x (3/2)

                     = 95/18


(4) If α and β are the roots of 3 x² - 6 x + 4 = 0, find the value of α² + β²

Solution:

α² + β² = (α + β)² - 2 α β

To get the values a,b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

 a = 3  b = - 6 and c = 4

Sum of roots α + β = -b/a

                           = - (-6)/3

                           = 6/3

                           = 2

Product of roots α β = c/a 

                            = 4/3

α² + β² = (α + β)² - 2 α β

           = (2)² - 2 (4/3)

           = 4 - (8/3)

           = (12 - 8)/3

           = 4/3        

These are the problems solved in the page relations between roots solution3. You can get solution of other problems in the next page.

relations between roots solution3