# 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 = 0Sum of roots = aα + β = a ----(1)Product of roots = b αβ = bβ = b/α ----(2) x2 - ex + f = 0Sum of roots = eα + α = e2α = e   α = e/2 ----(3)Product of roots = fα(α) = fα2 = 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.

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### SAT Math Videos

May 22, 24 06:32 AM

SAT Math Videos (Part 1 - No Calculator)

2. ### Simplifying Algebraic Expressions with Fractional Coefficients

May 17, 24 08:12 AM

Simplifying Algebraic Expressions with Fractional Coefficients