Nature of Roots Solution4





In this page nature of roots solution4 we are going to see solutions of some practice questions of the worksheet nature of roots.

(3) Show that the roots of the equation x²+ 2 (a + b) x + 2 (a² + b²) = 0 are unreal.

To prove that the roots of the equation are unreal we have to find the value of ∆.

    ∆ = b² - 4 a c

a = 1    b = 2 (a + b)       c = 2 (a² + b²)

    ∆ = [2 (a + b)]² - 4 (1) [2 (a² + b²)]

       = 4 (a + b)² - 8 (a² + b²)

       = 4 (a² + b² + 2 a b) - 8 a² - 8 b²

       = 4 a² + 4 b² + 8 a b - 8 a² - 8 b²

       = 4 a² - 8 a² + 4 b² - 8 b² + 8 a b 

       = - 4 a² - 4 b² + 8 a b 

       = - 4 (a² + b² - 2 a b)

       = - 4 (a - b)²

∆ < 0. Hence we can decide that the roots are unreal.


(4) Show that the roots of the equation 3 p² x² - 2 p q x + q² = 0 are not real.

To prove that the roots of the equation are real we have to find the value of ∆.

a = 3 p²   b = - 2 p q        c = q²

    ∆ = b² - 4 a c

    ∆ = [-2 p q ]² - 4 (3 p²) (q²)

       = 4 p² q² - 12 p² q²

       = - 8 p² q²

∆ < 0. Hence we can decide that the roots are not real.


(5) If the roots of the equation (a² + b²) x² - 2 (a c + b d) x + (c² +d²) = 0 where a , b , c and d not equal to zero,prove that (a/b) = (c/d) 

If the roots are equal then ∆ = 0

∆ = b² - 4 a c

a = (a² + b²)     b = - 2 (a c + b d)    c = (c² +d²)

 [- 2 (a c + b d)]² - 4 (a² + b²) (c² +d²)  = 0

4 [(a c)² + (b d)² + 2 (a c) (b d) - 4 [a² c² + a² d² + b² c² + b² d²] = 0

4 [a² c²+ b² d² + 2 (a c) (b d)] - 4 a² c² - 4 a² d² - 4 b² c² - 4 b² d² = 0

4 a²c²+ 4 b²d² + 8 (a c) (b d) - 4 a² c² - 4 a² d² - 4 b² c² - 4 b² d² = 0

4 a²c² - 4 a² c² + 4 b²d² - 4 b² d²+ 8 (a c) (b d) - 4 a² d² - 4 b² c² = 0

- 4 a² d² - 4 b² c² + 8 (a c) (b d) = 0

-4 [(ad)² + (bc)² - 2 (a d) (b c)] = 0

     -4 (a d - b c)² = 0

         (a d - b c)² = 0 

          a d - b c = 0

                a d = b c

                (a/b) = (c/d)

Hence proved           nature of roots solution4

nature of roots solution4 nature of roots solution4