"Nature of the roots of quadratic equations worksheet pdf" is nothing but the pdf document which contains questions and answers on the above mentioned stuff.

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**Example 1 :**

Examine the nature of 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. **

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 2 :**

Examine the nature of 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.**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 3 :**

Examine the nature of 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.**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 4 :**

Examine the nature of the 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. **

Now we are going to look at some quiet different problems on "nature of the roots of quadratic equations worksheets pdf"

**Example 5 :**

If the roots of the equation 2x² + 8x - m³ = 0 are equal , then find the value of "m" ** **

**Solution :**

If 2x² + 8x - m³ =0 is compared to the general form ax² + bx + c =0,

we get a = 2, b = 8 and c = -m³.

Since the roots are equal, we have

b² - 4ac = 0

8² - 4(2)(-m³) = 0

64 + 8m³ = 0

8m³ = -64

m³ = -8

m³ = (-2)³

m = - 2

**Hence, the value of "m" is "-2".**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 6 :**

If the roots of the equation x² - (p+4)x + 2p + 5 = 0 are equal , then find the value of "p" ** **

**Solution :**

If x² - (p+4)x + 2p + 5 = 0 is compared to the general form ax² + bx + c =0,

we get a = 1, b = - (p+4) and c = 2p+5

Since the roots are equal, we have

b² - 4ac = 0

[-(p+4)]² - 4(1)(2p+5) = 0

(p+4)² - 8p - 20 = 0

p² + 8p + 16 -8p -20 = 0

p² - 4 =0

p² = 4

p = ± 2

**Hence, the value of "p" is "**** ±2 ".**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 7 :**

If the roots of the equation x² + (2p-1)x + p² = 0 are real , then find the value of "p" ** **

**Solution :**

If x² + (2p-1)x + p² = 0 is compared to the general form ax² + bx + c =0,

we get a = 1, b = 2p-1 and c = p²

Since the roots are real, we have

b² - 4ac ≥ 0

(2p-1)² - 4(1)(p²) ≥ 0

4p² - 4p +1 -4p² ≥ 0

- 4p +1 ≥ 0

1 ≥ 4p (or) 4p ≤ 1

p ≤ 1/4

**Hence, the value of "p" is less than or equal to "1/4****".**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 8 :**

If the roots of the equation x² - 16x + k =0 are real and equal, then find the value of "k"

**Solution :**

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

we get a = 1, b = -16 and c = k

Since the roots are real, we have

b² - 4ac = 0

(-16)² - 4(1)(k) = 0

64 - 4k = 0

64 = 4k

16 = k

(or) k = 4

**Hence, the value of "k" is "4****".**

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 9 :**

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

**x² - 5x = 2(3x+1) **

**Solution :**

First, let us write the given equation in general form.

x² - 5x = 2(3x+1)

x² - 5x = 6x+2

x² - 11x -2 = 0

If x² -11x - 2 = 0 is compared to the general form ax² + bx + c =0,

we get a = 1, b = -11 and c = -2

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

b² - 4ac = (-11)² - 4(1)(-2)

b² - 4ac = 121 + 8

b² - 4ac = 129 (>0, but not a perfect square)

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

Let us look at the next example on "Nature of the roots of quadratic equations worksheet pdf"

**Example 10 :**

Examine the nature of the roots of the following quadratic equation. nature of the roots of quadratic equations worksheet p

**Solution :**

If 2x² - 9x -6 = 0 is compared to the general form ax² + bx + c =0,

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

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

b² - 4ac = (-9)² - 4(2)(-6)

b² - 4ac = 81+ 48

b² - 4ac = 129 (>0, but not a perfect square)

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

We hope that the student would have understood the problems and solutions given on "nature of the roots of quadratic equations worksheet pdf"**. **

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**Related Topics :**

**1. Nature of the roots of a quadratic equation (Detailed stuff)**

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