DISCUSS THE NATURE OF ROOTS OF A QUADRATIC EQUATION

Problem 1 :

If the equations x2 - ax + b = 0 and x2 - ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f).

Solution :

Let α be the common root for both quadratic equations

Let β be the other root of the quadratic equations

Since the roots of the second equation will be same, α and α are the roots of the second equation.

x2 - ax + b = 0

Sum of roots = a

α + β = a ----(1)

Product of roots = b

αβ = b

β = b/α ----(2)

x2 - ex + f = 0

Sum of roots = e

α + α = e

2α = e 

 α = e/2 ----(3)

Product of roots = f

α(α) = f

α2 = f ----(4)

Problem 2 :

Discuss the nature of roots of

−x2 + 3x + 1 = 0

Solution :

To find the nature of roots, we have to use the formula for discriminant.

Discriminant  =  b2 - 4ac

a  =  -1, b = -3 and c  =  1

  =  (-3)2 - 4(-1) (1)

  =  9 + 4

  =  13 > 0

So, the roots are real and distinct.

Problem 3 :

Discuss the nature of roots of

4x2 − x − 2 = 0

Solution :

To find the nature of roots, we have to use the formula for discriminant.

Discriminant  =  b2 - 4ac

a  =  4, b = -1 and c  =  -2

  =  (-1)2 - 4(4) (-2)

  =  1 + 64

  =  65 > 0

So, the roots are real and distinct.

Problem 4 :

Discuss the nature of roots of

9x2 + 5x = 0.

Solution :

To find the nature of roots, we have to use the formula for discriminant.

Discriminant  =  b2 - 4ac

a  =  9, b = 5 and c  =  0

  =  (5)2 - 4(9) (0)

  =  25 > 0

So, the roots are real and distinct.

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