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.

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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