# EXAMINE THE ROOTS OF A QUADRATIC EQUATION

## About "Examine the roots of a quadratic equation"

"Examine the roots of a quadratic equation" is nothing but the stuff which is "Nature of the roots of a quadratic equation".

To understand how to examine 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 examine the roots of a quadratic-equation, we will be using the discriminant **"b****² - 4ac"**.

Because **"b****² - 4ac"**discriminates the nature of the roots.

Let us see how this discriminant **"b****² - 4ac"** can be used to know the nature of the roots of a quadratic-equation.

## Examples

**Example 1 :**

Examine the roots of the following quadratic equation.

**x² + 5x + 6 =0 **

**Solution :**

If x² + 5x + 6 =0 is compared to the general form ax² + bx + c =0,

we get a = 1, b = 5 and c = 6.

Now, let us find the value of the discriminant "b² - 4ac"

b² - 4ac = 5² - 4(1)(6)

b² - 4ac = 25 - 24

b² - 4ac = 1 (>0 and also a perfect square)

**Hence, the roots are real, distinct and rational. **

**Example 2 :**

Examine the roots of the following quadratic-equation.

**2x² - 3x + 1 =0 **

**Solution :**

If 2x² - 3x + 1 =0 is compared to the general form ax² + bx + c =0,

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

Now, let us find the value of the discriminant "b² - 4ac"

b² - 4ac = (-3)² - 4(2)(-1)

b² - 4ac = 9 + 8

b² - 4ac = 17 (>0 but not a perfect square)

**Hence, the roots are real, distinct and irrational. **

**Example 3 :**

Examine the roots of the following quadratic-equation.

**x² - 16x + 64 =0 **

**Solution :**

If x² - 16x + 64 =0 is compared to the general form ax² + bx + c =0,

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

Now, let us find the value of the discriminant "b² - 4ac"

b² - 4ac = (-16)² - 4(1)(64)

b² - 4ac = 256 - 256

b² - 4ac = 0

**Hence, the roots are real, equal and rational. **

**Example 4 :**

Examine the of roots of the following quadratic-equation.

**3x² + 5x + 8 =0 **

**Solution :**

If 3x² + 5x + 8 =0 is compared to the general form ax² + bx + c =0,

we get a = 3, b = 5 and c = 8.

Now, let us find the value of the discriminant "b² - 4ac"

b² - 4ac = 5² - 4(3)(8)

b² - 4ac = 25- 96

b² - 4ac = -71 (negative)

**Hence, the roots are imaginary. **

Apart from the stuff and examples given above, if you want to know more about nature of the roots of a quadratic equation, please click here.