# NATURE OF THE ROOTS OF QUADRATIC EQUATIONS WORKSHEET PDF

"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.

If you want to practice the problems in the above pdf document, first you have to know the following stuff.

Now let us look at some examples on "nature of the roots of quadratic equations worksheet pdf"

## Examples

Now, let us see the step by step solutions for the problems you find in "Nature of the roots of quadratic equations worksheet pdf".

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 -4 ≥ 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".

Apart from "nature of the roots of quadratic equations worksheet pdf" examples, you can also visit the following web pages to know more about quadratic equation. Because learning the below stuff would be much helpful for you to do problems on "nature of the roots of  quadratic equations worksheet pdf

Related Topics  :

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

2. Relationship between zeros and coefficients of a quadratic polynomial

3. Word problems on quadratic equation