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

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relations between roots solution3