# DETERMINE THE NATURE OF THE ROOTS OF THE QUADRATIC EQUATION

## About "Determine the nature of the roots of the quadratic equation"

"Determine the nature of the roots of the quadratic equation" is the question being faced by all the students who study high school math.

To answer the above question, 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 **"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 :**

Determine the nature of the roots of the quadratic equation given below.

**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 :**

Determine the nature of the roots of the quadratic equation given below.

**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 :**

Determine the nature of the roots of the quadratic equation given below.

**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 :**

Determine the nature of the roots of the quadratic equation given below.

**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. **

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