NATURE OF THE ROOTS OF QUADRATIC EQUATIONS WORKSHEET PDF

About "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.

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Examples

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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. 

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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.

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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.

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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".

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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 ".

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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. 

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