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.

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.  

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