Relations Between Roots Solution7





The page relations between roots solution7 is containing solution of some practice questions from the worksheet relationship between roots and coefficients.

(9) Find a quadratic equation whose roots are the reciprocal of the roots of the equation 4 x² - 3 x - 1 = 0.

Solution:

General form of quadratic equation whose roots are α and β

 x² - (α + β) x + α β = 0

here we need to form a equation whose roots are reciprocal of the roots of the equation 4 x² - 3 x - 1 = 0.

by comparing the given equation with general form of quadratic equation we get a = 4  b = -3 and c = -1

Sum of the roots α + β = -b/a

                                = -(-3)/4

                                = 3/4

Product of roots α β = c/a

                             = -1/4

 here α = 1/α  and β = 1/β

General form of quadratic equation whose roots are 1/α and 1/β

 x² - (1/α + 1/β) x + (1/α) (1/β) = 0

 x² - [(α + β)/αβ] x + [(1/αβ)] = 0

 x² - [(3/4)/(-1/4)] x + [(1/(-1/4)] = 0

 x² - [(3/4)x(-4/1)] x + [(1/(-1/4)] = 0

 x² - (-3) x - 4 = 0

 x² + 3 x - 4 = 0

Therefore the required equation is x² + 3 x - 4 = 0.


(10) If one root of the equation 3 x² + k x - 81 = 0 is the square of the other, find k.

Solution:

Two roots of any quadratic equation are α and β. Here one root is square of the other

 α = β²

by comparing the given equation with general form of quadratic equation we get a = 3  b = k and c = -81

Sum of the roots α + β = -b/a

                                = -k/3

Product of roots α β = c/a

                             = -81/2

                             = -27


     α + β = -k/3

     β² + β = -k/3 ---- (1)

     α β = -27

     β²(β) = -27

     β³ = (-3)³

      β = - 3

now we are going to apply the value of β in the first equation

    β² + β = -k/3

    (-3)² + (-3) = k/3

      9 - 3 = k/3

        6 = k/3

         18 = k

Therefore the value of k is 18.

These are the problems solved in the page relations between roots solution7.

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relations between roots solution7