NATURE OF THE ROOTS OF A QUADRATIC EQUATION

About "Nature of the Roots of a Quadratic Equation"

Nature of the Roots of a Quadratic Equation :

In this section, we will learn about the nature of the roots of a quadratic equation.

That is, we will analyse whether the roots of a quadratic equation are equal or unequal,  real or imaginary and rational or irrational. 

To understand the nature of the roots of a quadratic equation, let us consider the general form a quadratic equation.

 ax² + bx + c = 0

(Here a, b and c are real and rational numbers)

To know the nature of the roots of a quadratic-equation, we will be using the discriminant b² - 4ac.

Because b² - 4ac discriminates the nature of the roots.

Let us see how this discriminant b² - 4ac can be used to know the nature of the roots of a quadratic-equation.

Nature of the Roots of a Quadratic Equation - Examples

Example 1 :

Examine the nature of the roots of the following quadratic equation.

3x² + 8x + 4  =  0

Solution :

The given quadratic equation is in the general form

ax² + bx + c  =  0

Then, we have a  =  3, b  =  8 and c  =  4.

Find the value of the discriminant b² - 4ac. 

b² - 4ac  =  8² - 4(3)(4)

b² - 4ac  =  64 - 48

b² - 4ac  =  16 

Here, b² - 4ac > 0 and also a perfect square. 

So, the roots are real, unequal and rational.  

Example 2 :

Examine the nature of the roots of the following quadratic equation.

2x² - 3x - 1  =  0 

Solution :

The given quadratic equation is in the general form

ax² + bx + c  =  0

Then, we have a  =  2, b  =  -3 and c  =  -1.

Find the value of the discriminant b² - 4ac. 

b² - 4ac  =  (-3)² - 4(2)(-1)

b² - 4ac  =  9 + 8

b² - 4ac  =  17 

Here, b² - 4ac > 0, but not a perfect square. 

So, the roots are real, unequal and irrational.  

Example 3 :

Examine the nature of the roots of the following quadratic equation.

x² - 16x + 64  =  0

Solution :

The given quadratic equation is in the general form

ax² + bx + c  =  0

Then, we have a  =  1, b  =  -16 and c  =  64.

Find the value of the discriminant b² - 4ac. 

b² - 4ac  =  (-16)² - 4(1)(64)

b² - 4ac  =  256 - 256

b² - 4ac  =  0 

So, the roots are real, equal and rational.  

Example 4 :

Examine the nature of the roots of the following quadratic equation.

5x² - 4x + 2  =  0

Solution :

The given quadratic equation is in the general form

ax² + bx + c  =  0

Then, we have a  =  5, b  =  -4 and c  =  2.

Find the value of the discriminant b² - 4ac. 

b² - 4ac  =  (-4)² - 4(5)(2)

b² - 4ac  =  16 - 40

b² - 4ac  =  -24 

Here, b² - 4ac < 0. 

So, the roots are imaginary. 

Problem 5 : 

Examine the nature of the roots of the following quadratic equation.

1/(x+1)  +  2/(x-4)  =  2

Solution :

The given quadratic equation is not in the general form.

First, write the given quadratic equation in the general form. 

1/(x+1)  +  2/(x-4)  =  2

Add the two fractions on the right side of the equation using cross multiplication. 

[(x-4) + 2(x+1)] / [(x+1)(x-4)]  =  2

(x - 4 + 2x + 2) / (x² - 3x - 4)  =  2

(3x - 2) / (x² - 3x - 4)  =  2

Multiply each side by (x² - 3x - 4). 

3x - 2  =  2(x² - 3x - 4)

3x - 2  =  2x² - 6x - 8

2x² - 9x - 6  =  0

Now, the quadratic equation is the general form

ax² + bx + c  =  0

Then, we have a  =  2, b  =  -9 and c  =  -6.

Find the value of the discriminant b² - 4ac. 

b² - 4ac  =  (-9)² - 4(2)(-6)

b² - 4ac  =  81 + 48

b² - 4ac  =  129

Here, b² - 4ac > 0, but not a perfect square. 

So, the roots are real, unequal and irrational.  

After having gone through the stuff given above, we hope that the students would have understood, "Nature of the Roots of a Quadratic Equation". 

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